IKEIEY 

RARY 

ERSITY  Of 
IIFORNIA 


[TECTUBE  LIB. 


/i   HISTORY   OF   MATHEMATICS 


A  HISTORY  OF 

MATHEMATICS 


BY 
FLORIAN   CAJORI,    Ph.D. 

PROFESSOR  OF  HISTORY  OF  MATHEMATICS  IN  THE 
UNIVERSITY  OF  CALIFORNIA 


"  I  am  sure  that  no  subject  loses  more  than  mathematicf 
by  any  attempt  to  dissociate  it  from  its  history." — J.  W.  L 
Glaisher 


SECOND  EDITION,  REVISED   AND   ENLARGED 


THE  MACMILLAN  COMPANY 


AECHIIECIUBE  LIB, 


Copyright,  1893, 

By   MACMILLAN   AND   CO. 

Set  up  and  electrotyped  January.  I'^g;.  Reprinted  March,  1895: 
October,  1S97;  November,  igoi;  January,  1906;  July,  1905 
"sbruary,  191 1. 


Second  Edition,  Revised  and  Enlarged 

Copyright,  1919. 
By  the   MACMILLAN   COMPANY. 


All  rights  reserved  —  no  part  of  this 

book  may  be  reproduced  in  any  form 

without  permission  in  writing  from 

the  publisher. 


Ninth  Printing,  1955 


PRINTED  IN  THE   UNITED   STATES   OF  AMERICA 


PREFACE  TO  THE  SECOND  EDITION     "^  '  ^ 

ARCH. 

In  preparing  this  second  edition  the  earlier  portions  of  the  book 
have  been  partly  re-written,  while  the  chapters  on  recent  mathematics 
are  greatly  enlarged  and  almost  wholly  new.  The  desirability  of 
having  a  rehable  one- volume  history  for  the  use  of  readers  who  cannot 
devote  themselves  to  an  intensive  study  of  the  history  of  mathematics 
is  generally  recognized.  On  the  other  hand,  it  is  a  difficult  task  to 
give  an  adequate  bird's-eye-view  of  the  development  of  mathematics 
from  its  earliest  beginnings  to  the  present  time.  In  compiling  this 
history  the  endeavor  has  been  to  use  only  the  most  reliable  sources. 
Nevertheless,  in  covering  such  a  wide  territory,  mistakes  are  sure  to 
have  crept  in.  References  to  the  sources  used  in  the  revision  are 
given  as  fully  as  the  limitations  of  space  would  permit.  These  ref- 
erences will  assist  the  reader  in  following  into  greater  detail  the  his- 
tory of  any  special  subject.  Frequent  use  without  acknowledgment 
has  been  made  of  the  following  pubHcations:  Annuario  Biografico  del 
Circolo  Matematico  di  Palermo,  1914;  Jahrbuch  uber  die  Fortschritte  der 
MathcmaUk,  Berlin;  /.  C.  Poggendorff's  Biographisch-Literarisches 
Ha)idwdrterbuch,  Leipzig;  Gedenktagebuch  fur  Mathematiker,  von  Felix 
Miiller,  3.  Aufl.,  Leipzig  und  Berlin,  1^12;  Revue  Semestrielle  des  Pub- 
lications Mathematiqiies,  Amsterdam. 

The  author  is  indebted  to  Miss  Falka  M.  Gibson  of  Oakland,  Cal. 
for  assistance  in  the  reading  of  the  proofs. 

Flokian  Cajori. 

University  of  California., 
March,  191 9. 


7^3 


TABLE  OF   CONTENTS 

Page 

Introduction i 

The  Babylonians 4 

The  Egyptians 9 

The  Greeks 15 

Greek  Geometry 15 

The  Ionic  School 15 

The  School  of  Pythagoras 17 

The  Sophist  School 20 

The  Platonic  School 25 

The  First  Alexandrian  School 29 

The  Second  Alexandrian  School 45 

Greek  Arithmetic  and  Algebra 52 

The  Romans 63 

The  Maya 69 

The  Chinese 71 

The  Japanese 78 

The  Hindus 83 

The  Arabs 99 

Europe  during  the  Middle  Ages 113 

Introdiiction  of  Roman  Mathematics 113 

Translation  of  Arabic  Manuscripts 1 18 

The  First  Awakening  and  its  Sequel 120 

Europe  during  the  Sixteenth,  Seventeenth  and  Eight- 
eenth Centuries 130 

The  Renaissance 130 

Vieta  and  Descartes 145 

Descartes  to  Newton 173 

Newton  to  Enter 190 

Elder,  Lagrange  and  Laplace 231 

The  Nineteenth  and  Twentieth  Centuries 27s 

Introduction 278 

Definition  uf  Mathematics 285 

vii 


viii  CONTENTS 

Page 

Synthetic  Geometry 286 

Elementary  Geometry  of  the  Triangle  and  Circle 297 

Link-motion 301 

Parallel  Lines,  Non-Euclidean  Geometry  and  Geometry  of 

n  Dimensions 302 

Analytic  Geometry 309 

Analysis  Situs 323 

Intrinsic  Co-ordinates 324 

Definition  of  a  Curve 325 

Fundamental  Postulates 326 

Geometric  Models 328 

Algebra 329 

Theory  of  Equations  and  Theory  of  Groups 349 

Solution  of  Numerical  Equations _ 363 

Magic  Squares  and  Combinatory  Analysis 316 

A  nalysis 3^7 

Calculus  of  Variations 369 

Convergence  of  Series 373 

Probability  and  Statistics 377 

Differential  Equations.     Difference  Equations 383 

Integral  Equations,  Integro-differential  Equations,  General 

Analysis,  Functional  Calculus 392 

Theories  of  Irrationals  and  Theory  of  Aggregates 396 

Mathematical  Logic 407 

Theory  of  Functions 411 

Elliptic  Functions 411 

General  Theory  of  Functions 419 

Uniformization 433 

Theory  of  Numbers 434 

Fermat's  "Last  Theorem,"  Waring's  Theorem 442 

Other  Recent  Researches.    Number  Fields 444 

Transcendental  Numbers.    The  Infinite 446 

Applied  Mathematics 447 

Celestial  IVIechanics 447 

Problem  of  Three  Boc^ies 452 

General  Mechanics 455 

Fluid  Motion 460 

Sound.    Elasticity 464 

Light,  Electricity,  Heat,  Potential 470 

Relativity 479 

Nomography 481 

Mathematical  Tables 482 

Calculating  Machines,  Planimeters,  Integraphs 485 

Alphabetical  Index 487 


A  HISTORY   OF   MATHEMATICS 


A   HISTORY   OF  MATHEMATICS 


INTRODUCTION 

The  contemplation  of  the  various  steps  by  which  mankind  has 
come  into  possession  of  the  vast  stock  of  mathematical  knowledge 
can  hardly  fail  to  interest  the  mathematician.  He  takes  pride  in  the 
fact  that  his  science,  more  than  any  other,  is  an  exact  science,  and 
that  hardly  anything  ever  done  in  mathematics  has  proved  to  be  use- 
less. The  chemist  smiles  at  the  childish  efforts  of  alchemists,  but  the 
mathematician  finds  the  geometry  of  the  Greeks  and  the  arithmetic 
of  the  Hindus  as  useful  and  admirable  as  any  research  of  to-day.  He 
is  pleased  to  notice  that  though,  in  course  of  its  development,  mathe- 
matics has  had  periods  of  slow  growth,  yet  in  the  main  it  has  been 
pre-eminently  a  progressive  science. 

The  history  of  mathematics  may  be  instructive  as  well  as  agreeable; 
H  may  not  only  remind  us  of  what  we  have,  but  may  also  teach  us 
how  to  increase  our  store.  Says  A.  De  Morgan,  "The  early  history 
of  the  mind  of  men  with  regard  to  mathematics  leads  us  to  point  out 
our  own  errors;  and  in  this  respect  it  is  well  to  pay  attention  to  the 
history  of  mathematics."  It  warns  us  against  hasty  conclusions;  it 
points  out  the  importance  of  a  good  notation  upon  the  progress  of  the 
science;  it  discourages  excessive  specialisation  on  the  part  of  investi- 
gators, by  showing  how  apparently  distinct  branches  have  been  found 
to  possess  unexpected  connecting  links;  it  saves  the  student  from 
wasting  time  and  energy  upon  problems  which  were,  perhaps,  solved 
long  since;  it  discourages  him  from  attacking  an  unsolved  problem  by 
the  same  method  which  has  led  other  mathematicians  to  failure;  it 
teaches  that  fortifications  can  be  taken  in  other  ways  than  by  direct 
attack,  that  when  repulsed  from  a  direct  assault  it  is  well  to  recon- 
noitre and  occupy  the  surrounding  ground  and  to  discover  the  secret 
paths  by  which  the  apparently  unconquerable  position  can  be  taken.  ^ 
The  importance  of  this  strategic  rule  may  be  emphasised  by  citing  a 
case  in  which  it  has  been  violated.  An  untold  amount  of  intellectual 
energy  has  been  expended  on  the  quadrature  of  the  circle,  yet  no  con- 
quest has  been  made  by  direct  assault.  The  circle-squarers  have 
existed  in  crowds  ever  since  the  period  of  Archimedes.  After  in- 
numerable failures  to  solve  the  problem  at  a  time,  even,  when  in- 

*  S.  Giinther,  Ziele  und  ResuUate  der  neueren  Mathematisch-historischen  Forschung. 
Erlangen,  1876. 

I 


2  A  HISTORY  OF  MATHEMATICS 

vestigators  possessed  that  most  powerful  tool,  the  diflferential  calculus, 
persons  versed  in  mathematics  dropped  the  subject,  while  those  who 
still  persisted  were  completely  ignorant  of  its  history  and  generally 
misunderstood  the  conditions  of  the  problem.  "Our  problem,"  says 
A.  De  Morgan,  "  is  to  square  the  circle  with  the  old  allowance  of  means: 
Euclid's  postulates  and  nothing  more.  We  cannot  remember  an 
instance  in  which  a  question  to  be  solved  by  a  definite  method  was 
tried  by  the  best  heads,  and  answered  at  last,  by  that  method,  after 
thousands  of  complete  failures."  But  progress  was  made  on  this 
problem  by  approaching  it  from  a  different  direction  and  by  newly 
discovered  paths.  J.  H.  Lambert  proved  in  1761  that  the  ratio  of  the 
circumference  of  a  circle  to  its  diameter  is  irrational.  Some  years 
ago,  F.  Lindemann  demonstrated  that  this  ratio  is  also  transcendental 
and  that  the  quadrature  of  the  circle,  by  means  of  the  ruler  and  com- 
passes only,  is  impossible.  He  thus  showed  by  actual  proof  that  which 
keen-minded  mathematicians  had  long  suspected;  namely,  that  the 
great  army  of  circle-squarers  have,  for  two  thousand  years,  been 
assaulting  a  fortification  which  is  as  indestructible  as  the  firmament 
of  heaven. 

Another  reason  for  the  desirability  of  historical  study  is  the  value 
of  historical  knowledge  to  the  teacher  of  mathematics.  The  interest 
which  pupils  take  in  their  studies  may  be  greatly  increased  if  the 
solution  of  problems  and  the  cold  logic  of  geometrical  demonstrations 
are  interspersed  with  historical  remarks  and  anecdotes.  A  class  in 
arithmetic  vvill  be  pleased  to  hear  about  the  Babylonians  and  Hindus 
and  their  invention  of  the  "Arabic  notation";  they  will  marvel  at 
the  thousands  of  years  which  elapsed  before  people  had  even  thought 
of  introducing  into  the  numeral  notation  that  Columbus-egg — the 
zero;  they  will  find  it  astounding  that  it  should  have  taken  so  long 
to  invent  a  notation  which  they  themselves  can  now  learn  in  a  month. 
After  the  pupils  have  learned  how  to  bisect  a  given  angle,  surprise 
them  by  telling  of  the  many  futile  attempts  which  have  been  made 
to  solve,  by  elementary  geometry,  the  apparently  very  simple  problem 
of  the  trisection  of  an  angle.  When  they  know  how  to  construct  a 
square  whose  area  is  double  the  area  of  a  given  square,  tell  them  about 
the  duplication  of  the  cube,  of  its  mythical  origin — how  the  wrath  of 
Apollo  could  be  appeased  only  by  the  construction  of  a  cubical  altar 
double  the  given  altar,  and  how  mathematicians  long  wrestled  with 
this  problem.  After  the  class  have  exhausted  their  energies  on  the 
theorem  of  the  right  triangle,  tell  them  the  legend  about  its  discov- 
erer— how  Pythagoras,  jubilant  over  his  great  accomplishment, 
sacrificed  a  hecatomb  to  the  Muses  who  inspired  him.  When  the 
value  of  mathematical  training  is  called  in  question,  quote  the  in- 
scription over  the  entrance  into  the  academy  of  Plato,  the  philosopher: 
"Let  no  one  who  is  unacquainted  with  geometry  enter  here."  Students 
in  analytical  geometry  should  know  something  of  Descartes,  and, 


INTRODUCTION  5 

after  taking  up  the  differential  and  integral  calculus,  they  should 
become  familiar  with  the  parts  that  Newton,  Leibniz,  and  Lagrange 
played  in  creating  that  science.  In  his  historical  talk  it  is  possible 
for  the  teacher  to  make  it  plain  to  the  student  that  mathematics  is 
not  a  dead  science,  but  a  living  one  in  which  steady  progress  is  made.^ 

A  similar  point  of  view  is  taken  by  Henry  S.  White:  -  ''The  ac- 
cepted truths  of  to-day,  even  the  commonplace  truths  of  any  science, 
were  the  doubtful  or  the  novel  theories  of  yesterday.  Some  indeed 
of  prime  importance  were  long  esteemed  of  slight  importance  and 
almost  forgotten.  The  first  effect  of  reading  in  the  history  of  science 
is  a  naive  astonishment  at  the  darkness  of  past  centuries,  but  the 
ultimate  effect  is  a  fervent  admiration  for  the  progress  achieved  by 
former  generations,  for  the  triumphs  of  persistence  and  of  genius. 
The  easy  credulity  with  which  a  young  student  supposes  that  of 
course  every  algebraic  equation  must  have  a  root  gives  place  finally 
to  a  delight  in  the  slow  conquest  of  the  realm  of  imaginary  numbers, 
and  in  the  youthful  genius  of  a  Gauss  who  could  demonstrate  this 
once  obscure  fundamental  proposition." 

The  history  of  mathematics  is  important  also  as  a  valuable  con- 
tribution to  the  history  of  civilisation.  Human  progress  is  closely 
identified  with  scientific  thought.  Mathematical  and  physical  re- 
searches are  a  reliable  record  of  intellectual  progress.  The  history 
of  mathematics  is  one  of  the  large  windows  through  which  the  philo- 
sophic eye  looks  into  past  ages  and  traces  the  line  of  intellectual  de- 
velopment. 

*  Cajori,  F.,  The  Teaching  and  History  of  Mathematics  in  the  United  States.  Wash- 
ington, 1890,  p.  236. 

^Bidl.  Am.  Math.  Soc,  Vol.  15,  1909,  p.  325. 


THE  BABYLONIANS 

The  fertile  valley  of  the  Euphrates  and  Tigris  was  one  of  the 
primeval  seats  of  human  society.  Authentic  history  of  the  peoples 
inhabiting  this  region  begins  only  with  the  foundation,  in  Chaldaea 
and  Babylonia,  of  a  united  kingdom  out  of  the  previously  disunited 
tribes.  Much  light  has  been  thrown  on  their  history  by  the  discovery 
of  the  art  of  reading  the  cuneiform  or  wedge-shaped  system  of  writing. 

In  the  study  of  Babylonian  mathematics  we  begin  with  the  notation 
of  numbers.  A  vertical  wedge  ]f  stood  for  i,  while  the  characters  .^ 
and  Y^*  signified  lo  and  loo  respectively.  Grotefend  believes  the 
character  for  lo  originally  to  have  been  the  picture  of  two  hands,  as 
held  in  prayer,  the  palms  being  pressed  together,  the  fingers  close  to 
each  other,  but  the  thumbs  thrust  out.  In  the  Babylonian  notation 
two  principles  were  employed — the  additive  and  multiplicative.  Num- 
bers below  ICO  were  expressed  by  symbols  whose  respective  values 
had  to  be  added.  Thus,  \  \  stood  for  2,  f  J  f  for  3,  X^I  for  4,<t* 
for  23,  ^  ^  ^  for  30.  Here  the  symbols  of  higher  order  appear 
always  to  the  left  of  those  of  lower  order.  In  writing  the  hun- 
dreds, on  the  other  hand,  a  smaller  symbol  was  placed  to  the  left  of 
100,  and  was,  in  that  case,  to  be  multiplied  by  100.  Thus,  ^  y  ^ 
signified  10  times  100,  or  1000.  But  this  symbol  for  1000  was  itself 
taken  for  a  new  unit,  which  could  take  smaller  coefficients  to  its  left. 
Thus,  ^  ^  y  ^  denoted,  not  20  times  100,  but  10  times  1000.  Some 
of  the  cuneiform  numbers  found  on  tablets  in  the  ancient  temple 
library  at  Nippur  exceed  a  million;  moreover,  some  of  these  Nippur 
tablets  exhibit  the  subtractive  principle  (20-1),  similar  to  that  shown 
in  the  Roman  notation,  "XIX." 

If,  as  is  believed  by  most  specialists,  the  early  Sumerians  were  the 
inventors  of  the  cuneiform  writing,  then  they  were,  in  all  probabiUty, 
also  the  inventors  of  the  notation  of  numbers.  Most  surprising,  in 
this  connection,  is  the  fact  that  Sumerian  inscriptions  disclose  the  use, 
not  only  of  the  above  decimal  system,  but  also  of  a  sexagesimal  one. 
The  latter  was  used  chiefly  in  constructing  tables  for  weights  and 
measures.  It  is  full  of  historical  interest.  Its  consequential  develop- 
ment, both  for  integers  and  fractions,  reveals  a  high  degree  of  mathe- 
matical insight.  We  possess  two  Babylonian  tablets  which  exhibit 
its  use.  One  of  them,  probably  written  between  2300  and  1600  b.  c, 
contains  a  table  of  square  numbers  up  to  60^.  The  numbers  i,  4,  9, 
16,  25,  36,  49,  are  given  as  the  squares  of  the  first  seven  integers  re- 

4 


THE  BAB\XONIANS  5 

spectively.  We  have  next  1.4  =  8',  1.21  =  9-,  1.40=10',  2.1  =  11',  etc. 
This  remains  unintelligible,  unless  we  assume  the  sexagesimal  scale, 
which  makes  1.4=60+4,  1.21  =  60+21,  2.1  =  2.60+1.  The  second 
tablet  records  the  magnitude  of  the  illuminated  portion  of  the  moon's 
disc  for  every  day  from  new  to  full  moon,  the  whole  disc  being  assumed 
to  consist  of  240  parts.  The  illuminated  parts  during  the  first  five 
days  are  the  series  5,  10,  20,  40,  1.20  (=80),  which  is  a  geometrical 
progression.  From  here  on  the  series  becomes  an  arithmetical  progres- 
sion, the  numbers  from  the  fifth  to  the  fifteenth  day  being  respectively 
1.20,  1.36,  1.52,  2.8,  2.24,  2.40,  2.56,  3.12,  3. 28,  3.44,  4.  This  table 
not  only  exhibits  the  use  of  the  sexagesimal  system,  but  also  indicates 
the  acquaintance  of  the  Babylonians  with  progressions.  Not  to  be 
overlooked  is  the  fact  that  in  the  sexagesimal  notation  of  integers 
the  "principle  of  position"  was  employed.  Thus,  in  1.4  (=64),  the 
I  is  made  to  stand  for  60,  the  unit  of  the  second  order,  by  ^'irtue  of 
its  position  with  respect  to  the  4.  The  introduction  of  this  principle 
at  so  early  a  date  is  the  more  remarkable,  because  in  the  decimal  no- 
tation it  was  not  regularly  introduced  till  about  the  ninth  century 
after  Christ.  The  principle  of  position,  in  its  general  and  systematic 
application,  requires  a  symbol  for  zero.  We  ask.  Did  the  Babylonians 
possess  one?  Had  they  already  taken  the  gigantic  step  of  representing 
by  a  s>Tnbol  the  absence  of  units?  Neither  of  the  above  tables  answers 
this  question,  for  they  happen  to  contain  no  number  in  which  there 
was  occasion  to  use  a  zero.  Babylonian  records  of  many  centuries 
later — of  about  200  B.  C. — give  a  symbol  for  zero  which  denoted  the 
absence  of  a  figure  but  apparently  was  not  used  in  calculation.  It 
consisted  of  two  angular  marks  ^  one  above  the  other,  roughly  re- 
sembling two  dots,  hastily  written.  About  130  A.  D..  Ptolemy  in 
Alexandria  used  in  his  Almagest  the  Babylonian  sexagesimal  fractions, 
and  also  the  omicron  o  to  represent  blanks  in  the  sexagesimal  numbers. 
This  o  was  not  used  as  a  regular  zero.  It  appears  therefore  that  the 
Babylonians  had  the  principle  of  local  value,  and  also  a  symbol  for 
zero,  to  indicate  the  absence  of  a  figure,  but  did  not  use  this  zero  in 
computation.  Their  sexagesimal  fractions  were  introduced  into  India 
and  with  these  fractions  probably  passed  the  principle  of  local  value 
and  the  restricted  use  of  the  zero. 

The  sexagesimal  system  was  used  also  in  fractions.  Thus,  in  the 
Babylonian  inscriptions,  \  and  \  are  designated  by  30  and  20,  the 
reader  being  ex-pected,  in  his  mind,  to  supply  the  word  "sixtieths." 
The  astronomer  Hipparchus,  the  geometer  H^-psicles  and  the  as- 
tronomer Ptolemy  borrowed  the  sexagesimal  notation  of  fractions 
from  the  Babylonians  and  introduced  it  into  Greece.  From  that  time 
sexagesimal  fractions  held  almost  full  sway  in  astronomical  and  mathe- 
matical calculations  until  the  sixteenth  century,  when  they  finally 
yielded  their  place  to  the  decimal  fractions.  It  may  be  asked.  What 
led  to  the  invention  of  the  sexagesinial  system?    Why  was  it  that  60 


6  A  HISTORY  OF  MATHEMATICS 

parts  were  selected?  To  this  we  have  no  positive  answer.  Ten  was 
chosen,  in  the  decimal  system,  because  it  represents  the  number  of 
fingers.  But  nothing  of  the  human  body  could  have  suggested  60. 
Did  the  system  have  an  astronomical  origin?  It  was  supposed  that 
the  early  Babylonians  reckoned  the  year  at  360  days,  that  this  led 
to  the  division  of  the  circle  into  360  degrees,  each  degree  representing 
the  daily  amount  of  the  supposed  yearly  revolution  of  the  sun  around 
the  earth.  Now  they  were,  very  probably,  familiar  with  the  fact 
that  the  radius  can  be  applied  to  its  circumference  as  a  chord  6  times, 
and  that  each  of  these  chords  subtends  an  arc  measuring  exactly  60 
degrees.  Fixing  their  attention  upon  these  degrees,  the  division  into 
60  parts  may  have  suggested  itself  to  them.  Thus,  when  greater  pre- 
cision necessitated  a  subdivision  of  the  degree,  it  was  partitioned  into 
60  minutes.  In  this  way  the  sexagesimal  notation  was  at  one  time 
supposed  to  have  originated.  But  it  now  appears  that  the  Babylonians 
very  early  knew  that  the  year  exceeded  360  days.  Moreover,  it  is 
highly  improbable  that  a  higher  unit  of  360  was  chosen  first,  and  a 
lower  unit  of  60  afterward.  The  normal  development  of  a  number 
system  is  from  lower  to  higher  units.  Another  guess  is  that  the 
sexagesimal  system  arose  as  a  mixture  of  two  earlier  systems  of  the 
bases  6  and  10.^  Certain  it  is  that  the  sexagesimal  system  became 
closely  interwoven  with  astronomical  and  geometrical  science.  The 
division  of  the  day  into  24  hours,  and  of  the  hour  into  minutes  and 
seconds  on  the  scale  of  60,  is  attributed  to  the  Babylonians.  There  is 
strong  evidence  for  the  belief  that  they  had  also  a  division  of  the  day 
into  60  hours.  The  employment  of  a  sexagesimal  division  in  numeral 
notation,  in  fractions,  in  angular  as  well  as  in  time  measurement,  in- 
dicated a  beautiful  harmony  which  was  not  disturbed  for  thousands 
of  years  until  Hindu  and  Arabic  astronomers  began  to  use  sines  and 
cosines  in  place  of  parts  of  chords,  thereby  forcing  the  right  angle  to 
the  front  as  a  new  angular  unit,  which,  for  consistency,  should  have 
been  subdivided  sexagesimally,  but  was  not  actually  so  divided. 

It  appears  that  the  people  in  the  Tigro-Euphrates  basin  had  made 
very  creditable  advance  in  arithmetic.  Their  knowledge  of  arith- 
metical and  geometrical  progressions  has  already  been  alluded  to. 
lamblichus  attributes  to  them  also  a  knowledge  of  proportion,  and 
even  the  invention  of  the  so-called  musical  proportion.    Though  we 

1  M.  Cantor,  Vorlesungen  uber  Geschickte  der  Malheniatik,  i.  Bd.,  3.  Aufl.,  Leipzig, 
1907,  p.  37.  This  work  has  appeared  in  four  large  volumes  and  carries  the  history 
down  to  1799.  The  fourth  volume  (1908)  was  written  with  the  cooperation  of  nine 
scholars  from  Germany,  Italy,  Russia  and  the  United  States.  Moritz  Cantor  (1829- 
)  ranks  as  the  foremost  general  writer  of  the  nineteenth  century  on  the  history 
of  mathematics.  Born  in  Mannheim,  a  student  at  Heidelberg,  at  Gottingen  under 
Gauss  and  Weber,  at  Berlin  under  Uirichlet,  he  lectured  at  Heidelberg  where  in 
1877  he  became  ordinary  honorary  professor.  His  first  historical  article  was 
brought  out  in  1856,  but  not  until  1880  did  the  first  volume  of  his  well-known  history 
appear. 


THE  BABYLONIANS  7 

possess  no  conclusive  proof,  we  have  nevertheless  reason  to  believe 
that  in  practical  calculation  they  used  the  abacus.  Among  the  races 
of  middle  Asia,  even  as  far  as  China,  the  abacus  is  as  old  as  fable. 
Now,  Babylon  was  once  a  great  commercial  centre, — the  metropolis  of 
many  nations, — and  it  is,  therefore,  not  unreasonable  to  suppose  that 
her  merchants  employed  this  most  improved  aid  to  calculation. 

In  1S89  H.  V.  Hilprccht  began  to  make  excavations  at  Nuffar  (the 
ancient  Nippur)  and  found  brick  tablets  containing  multiplication  and 
division  tables,  tables  of  squares  and  square  roots,  a  geometric  progres- 
sion and  a  few  computations.  He  published  an  account  of  his  findings 
in  1906.^ 

The  divisions  in  one  tablet  contain  results  like  these:  "60^  divided 
by  2  =  6,480,000  each,"  "6o'*  divided  by  3  =  4,320,000  each,"  and 
so  on,  using  the  divisors  2,  3,  4,  5,  6,  8,  9,  10,  12,  15,  16,  18.  The  very 
first  division  on  the  tablet  is  interpreted  to  be  "60'*  divided  by  i>^  = 
8,640,000."  This  strange  appearance  of  f  as  a  divisor  is  difficult  to 
explain.  Perhaps  there  is  here  a  use  of  f  corresponding  to  the  Egyptian 
use  of  I  as  found  in  the  Ahmes  papyrus  at  a,  perhaps,  contemporaneous 
period.  It  is  noteworthy  that  60^=  12,960,000,  which  Hilprecht  found 
in  the  Nippur  brick  text-books,  is  nothing  less  than  the  mystic  "  Platonic 
number,"  the  "lord  of  better  and  worse  births,"  mentioned  in  Plato's 
Republic.  Most  probably,  Plato  received  the  number  from  the 
P}'thagoreans,  and  the  Pythagoreans  from  the  Babylonians.- 

In  geometry  the  Babylonians  accomplished  little.  Besides  the  divi- 
sion of  the  circumference  into  6  parts  by  its  radius,  and  into  360  de- 
grees, they  had  some  knowledge  of  geometrical  figures,  such  as  the 
triangle  and  quadrangle,  which  they  used  in  their  auguries.  Like  the 
Hebrews  (i  Kin.  7:23),  they  took  7r  =  3.  Of  geometrical  demonstra- 
tions there  is,  of  course,  no  trace.  "As  a  rule,  in  the  Oriental  mind 
the  intuitive  powers  eclipse  the  severely  rational  and  logical." 

Hilprecht  concluded  from  his  studies  that  the  Babylonians  pos- 
sessed the  rules  for  finding  the  areas  of  squares,  rectangles,  right  tri- 
angles, and  trapezoids. 

The  astronomy  of  the  Babylonians  has  attracted  much  attention. 
They  worshipped  the  heavenly  bodies  from  the  earliest  historic  times. 
When  Alexander  the  Great,  after  the  battle  of  Arbela  (331  b.  c),  took 
possession  of  Babylon,  Callisthenes  found  there  on  burned  brick  as- 
tronomical records  reaching  back  as  far  as  2234  b.  c.  Porphyrius  says 
that  these  were  sent  to  Aristotle.  Ptolemy,  the  Alexandrian  astrono- 
mer, possessed  a  Babylonian  record  of  eclipses  going  back  to  747  b.  c. 

1  Mathemalical,  Mdrological  and  Chronological  Tabids  from  the  Temple  Library 
of  Nippur,  by  H.  V.  Hilprecht.  Vol.  XX,  part  I,  Series  A,  Cuneiform  Texts,  pub- 
lished by  the  Babylonian  Expedition  of  the  University  of  Pennsylvania,  1906.  Con- 
sult also  D.  E.  Smith  in  Bull.  Am.  Math.  Soc.,  Vol.  13,  igoy,  p.  392. 

2 On  the  "Platonic  number"  consult  P.  Tannery  in  Revue  philosophiquc,  Vol.  I, 
1876,  p.  170;  Vol.  XIII,  1881,  p.  210;  Vol.  XV,  1883,  p.  573.  Also  G.  Loria  in  /.« 
scknze  esatte  ncWantica  grecia,  2  Ed.,  Milano,  1914,  AppencUce. 


8  A  HISTORY  OF  MATHEMATICS 

Epping  and  Strassmaier  '  have  thrown  considerable  light  on  Babylon- 
ian chronology  and  astronomy  by  explaining  two  calendars  of  the 
years  123  b.  c.  and  iii  b.  c,  taken  from  cuneiform  tablets  coming, 
presumably,  from,  an  old  observatory.  These  scholars  have  succeeded 
in  giving  an  account  of  the  Babylonian  calculation  of  the  new  and 
full  moon,  and  have  identified  by  calculations  the  Babylonian  names 
of  the  planets,  and  of  the  twelve  zodiacal  signs  and  twenty-eight 
normal  stars  which  correspond  to  some  extent  with  the  twenty-eight 
nakshatras  of  the  Hindus.  We  append  part  of  an  Assyrian  astronomical 
report,  as  translated  by  Oppert: — 

"To  the  King,  my  lord,  thy  faithful  servant,  Mai-Istar." 

"...  On  the  first  day,  as  the  new  moon's  day  of  the  month  Thammuz 
declined,  the  moon  was  again  visible  over  the  planet  Mercury,  as  I  had 
already  predicted  to  my  master  the  King.    I  erred  not." 

1  Epping,  J.,  Aslronomisches  aus  Babylon.  Unter  Mitwirkung  von  P.  J.  R.  Strass- 
naier.    Freiburg,  1889. 


THE  EGYPTIANS 

Though  there  is  difference  of  opinion  regarding  the  antiquity  of 
Egyptian  civihsation,  yet  all  authorities  agree  in  the  statement  that, 
however  far  back  they  go,  they  find  no  uncivilised  state  of  society. 
"Menes,  the  first  king,  changes  the  course  of  the  Nile,  makes  a  great 
reservoir,  and  builds  the  temple  of  Phthah  at  Memphis."  The  Egyp- 
tians built  the  p}.Tamids  at  a  very  early  period.  Surely  a  people  en- 
gaging in  enterprises  of  such  magnitude  must  have  knovMi  something 
of  mathematics — at  least  of  practical  mathematics. 

All  Greek  wTiters  are  unanimous  in  ascribing,  without  envy,  to 
Egypt  the  priority  of  invention  in  the  mathematical  sciences.  Plato 
in  Phcednis  says:  "At  the  Egyptian  city  of  Naucratis  there  was  a 
famous  old  god  whose  name  was  Theuth;  the  bird  which  is  called  the 
Ibis  was  sacred  to  him,  and  he  was  the  inventor  of  many  arts,  such 
as  arithmetic  and  calculation  and  geometry  and  astronomy  and 
draughts  and  dice,  but  his  great  discovery  was  the  use  of  letters." 

Aristotle  says  that  mathematics  had  its  birth  in  Egypt,  because 
there  the  priestly  class  had  the  leisure  needful  for  the  study  of  it. 
Geometry,  in  particular,  is  said  by  Herodotus,  Diodorus,  Diogenes 
Laertius,  lamblichus,  and  other  ancient  writers  to  have  originated  in 
Egypt.^  In  Herodotus  we  find  this  (II.  c.  109):  "They  said  also  that 
this  king  [Sesostris]  divided  the  land  among  all  Egyptians  so  as  to 
give  each  one  a  quadrangle  of  equal  size  and  to  draw  from  each  his 
revenues,  by  imposing  a  tax  to  be  levied  yearly.  But  every  one  from 
whose  part  the  river  tore  away  anything,  had  to  go  to  him  and  notify 
what  had  happened;  he  then  sent  the  overseers,  who  had  to  measure 
out  by  how  much  the  land  had  become  smaller,  in  order  that  the 
owner  might  pay  on  what  was  left,  in  proportion  to  the  entire  tax 
imposed.  In  this  way,  it  appears  to  me,  geometry  originated,  which 
passed  thence  to  Hellas." 

We  abstain  from  introducing  additional  Greek  opinion  regarding 
Egyptian  mathematics,  or  from  indulging  in  wild  conjectures.  We 
rest  our  account  on  documentary  evidence.  A  hieratic  pap}Tus,  in- 
cluded in  the  Rhind  collection  of  the  British  Museum,  was  deciphered 
by  Eisenlohr  in  1877,  and  found  to  be  a  mathematical  manual  con- 
taining problems  in  arithmetic  and  geometry.  It  was  written  by 
Ahmes  some  time  before  1700  b.  c,  and  was  founded  on  an  older  work 
believed  by  Birch  to  date  back  as  far  as  3400  b.  c!    This  curious 

^  C.  A.  Bretschneider  Die  Geometrie  und  die  Geometer  vor  Euklides.  Leipzig,  1870, 
pp.  6-8.  Carl  Anton  Bretschneider  (1808-1878)  was  professor  at  the  Realg>'mna- 
■sium  at  Gotha  in  Thuringia. 

Q 


lo  A  HISTORY  OF  MATHEMATICS 

papyrus — the  most  ancient  mathematical  handbook  known  to  us — 
puts  us  at  once  in  contact  with  the  mathematical  thought  in  Egypt  of 
three  or  five  thousand  years  ago.  It  is  entitled  "Directions  for  ob- 
taining the  Knowledge  of  all  Dark  Things."  We  see  from  it  that  the 
Egyptians  cared  but  little  for  theoretical  results.  Theorems  are  not 
found  in  it  at  all.  It  contains  "hardly  any  general  rules  of  procedure, 
but  chiefly  mere  statements  of  results  intended  possibly  to  be  ex- 
plained by  a  teacher  to  his  pupils."  ^  In  geometry  the  forte  of  the 
Egyptians  lay  in  making  constructions  and  determining  areas.  The 
area  of  an  isosceles  triangle,  of  which  the  sides  measure  lo  khets  (a 
unit  of  length  equal  to  16.6  m.  by  one  guess  and  about  thrice  that 
amount  by  another  guess  ^)  and  the  base  4  khets,  was  erroneously  given 
as  20  square  khets,  or  half  the  product  of  the  base  by  one  side.  The 
area  of  an  isosceles  trapezoid  is  found,  similarly,  by  multiplying  half 
the  sum  of  the  parallel  sides  by  one  of  the  non-parallel  sides.  The 
area  of  a  circle  is  found  by  deducting  from  the  diameter  \  of  its  length 
and  squaring  the  remainder.  Here  tt  is  taken  =  (-y-)^  =  3. 1604...,  a 
very  fair  approximation.  The  papyrus  explains  also  such  problems 
as  these, — To  mark  out  in  the  field  a  right  triangle  whose  sides  are 
10  and  4  units;  or  a  trapezoid  whose  parallel  sides  are  6  and  4,  and 
the  non-parallel  sides  each  20  units. 

Some  problems  in  this  papyrus  seem  to  imply  a  rudimentary  knowl- 
edge of  proportion. 

The  base-lines  of  the  pyramids  run  north  and  south,  and  east  and 
west,  but  probably  only  the  lines  running  north  and  south  were  deter- 
mined by  astronomical  observations.  This,  coupled  with  the  fact 
that  the  word  harpedonapta,  applied  to  Eg}^ptian  geometers,  means 
"rope-stretchers,"  would  point  to  the  conclusion  that  the  Egyptian, 
like  the  Indian  and  Chinese  geometers,  constructed  a  right  triangle 
upon  a  given  line,  by  stretching  around  three  pegs  a  rope  consisting 
of  three  parts  in  the  ratios  3:4:5,  and  thus  forming  a  right  triangle.^ 
If  this  explanation  is  correct,  then  the  Eg}^tians  were  familiar,  2000 
years  b.  c,  with  the  well-known  property  of  the  right  triangle,  for 
the  special  case  at  least  when  the  sides  are  in  the  ratio  3:  4:  5- 

On  the  walls  of  the  celebrated  temple  of  Horus  at  Edfu  have  been 
found  hieroglyphics,  written  about  100  b.  c,  which  enumerate  the 
pieces  of  land  owned  by  the  priesthood,  and  give  their  areas.  The 
area  of  any  quadrilateral,  however  irregular,  is  there  found  by  the 
formula  -^-.^-.  Thus,  for  a  quadrangle  whose  opposite  sides 
are  5  and  8,  20  and  15,  is  given  the  area  ii3|-  \}    The  incorrect  for- 

^  James  Gow,  A  Short  History  of  Greek  Mathematics.    Cambridge,  1884,  p.  16. 

^  A.  Eisenlohr,  Ein  mathematisches  Ilandbuch  der  altcn  Aegyptcr,  2.  Ausgabe,  Leip- 
zig, 1897,  p.  103;  F.  L.  Griffith  in  Proceedings  of  the  Society  of  Biblical  Archceology, 
1891,  1894. 

^  M.  Cantor,  op.  cit.  Vol.  I,  3.  Aufl.,  1907,  p.  105. 

*  H.  Hankel,  Zur  Geschichte  der  Mathematik  in  Alterlhiim  mid  Mittelalter,  Leipzig, 
1874,  p.  86. 


THE  EGYPTIANS  ii 

mulag  of  Ahmes  of  3000  years  b.  c.  yield  generally  closer  approxima- 
lions  than  those  of  the  Edfu  inscriptions,  written  200  years  after 
Euclid! 

The  fact  that  the  geometry  of  the  Egyptians  consists  chiefly  of 
constructions,  goes  far  to  explain  certain  of  its  great  defects.  The 
Egvi^tians  failed  in  two  essential  points  without  which  a  science  of 
geometry,  in  the  true  sense  of  the  word,  cannot  exist.  In  the  first 
place,  they  failed  to  construct  a  rigorously  logical  system  of  geometry, 
resting  upon  a  few  axioms  and  postulates.  A  great  many  of  their 
rules,  especially  those  in  solid  geometry,  had  probably  not  been  proved 
at  all,  but  were  known  to  be  true  merely  from  observation  or  as  mat- 
ters of  fact.  The  second  great  defect  was  their  inability  to  bring  the 
numerous  special  cases  under  a  more  general  view,  and  thereby  to 
arrive  at  broader  and  more  fundamental  theorems.  Some  of  the 
simplest  geometrical  truths  were  divided  into  numberless  special  cases 
of  which  each  was  supposed  to  require  separate  treatment. 

Some  particulars  about  Egyptian  geometry  can  be  mentioned  more 
advantageously  in  connection  with  the  early  Greek  mathematicians 
who  came  to  the  Egyjotian  priests  for  instruction. 

An  insight  into  Egyptian  methods  of  numeration  was  obtained 
through  the  ingenious  deciphering  of  the  hieroglyphics  by  Champol- 
lion,  Young,  and  their  successors.  The  symbols  used  were  the  fol- 
lowing: ji  for  I,  f^  for  10,  (^  for  100,  ^  for  1000,  (f  for  10,000,  ^::> 
for  100,000,  ^  for  1,000,000,  O  for  10,000,000.^  The  symbol  for 
I  represents  a  vertical  staff;  that  for  10,000  a  pointing  finger;  that 
for  100,000  a  burbot;  that  for  1,000,000,  a  man  in  astonishment.  The 
significance  of  the  remaining  symbols  is  very  doubtful.  The  writing 
of  numbers  with  these  hieroglyphics  was  very  cumbrous.  The  unit 
symbol  of  each  order  was  repeated  as  many  times  as  there  were  units 
in  that  order.  The  principle  employed  was  the  additive.  Thus,  23 
was  written  n    H  I  fl  | 

Besides  the  hierogh^phics,  Egypt  possesses  the  hieratic  and  demotic 
writings,  but  for  want  of  space  we  pass  them  by. 

Herodotus  makes  an  important  statement  concerning  the  mode  of 
computing  among  the  Egyptians.  He  says  that  they  ^^ calculate  with 
pebbles  by  moving  the  hand  from  right  to  left,  while  the  Hellenes 
move  it  from  left  to  right."  Herein  we  recognise  again  that  instru- 
mental method  of  figuring  so  extensively  used  by  peoples  of  antiquity. 
The  Egyptians  used  the  decimal  scale.  Since,  in  figuring,  they  moved 
their  hands  horizontally,  it  seems  probable  that  they  used  ciphering- 
boards  with  vertical  columns.  In  each  column  there  must  have  been 
not  more  than  nine  pebbles,  for  ten  pebbles  would  be  equal  to  one 
pebble  in  the  column  next  to  the  left. 

1  M.  Cantor,  op.  cit.  Vol.  I,  3.  Aufl.,  1907,  p.  82. 


12  A  HISTORY  OF  MATHEMATICS 

The  Ahmes  papyrus  contains  interesting  information  on  the  way 
in  which  the  Egyptians  employed  fractions.  Their  methods  of  opera- 
tion were,  of  course,  radically  different  from  ours.  Fractions  were  a 
subject  of  very  great  difficulty  with  the  ancients.  Simultaneous 
changes  in  both  numerator  and  denominator  were  usually  avoided. 
In  manipulating  fractions  the  Babylonians  kept  the  denominators  (60) 
constant.  The  Romans  likewise  kept  them  constant,  but  equal  to  12. 
The  Egyptians  and  Greeks,  on  the  other  hand,  kept  the  numerators 
constant,  and  dealt  with  variable  denominators.  Ahmes  used  the 
term  "fraction"  in  a  restricted  sense,  for  he  applied  it  only  to  unit- 
fractions,  or  fractions  having  unity  for  the  numerator.  It  was  desig- 
nated by  writing  the  denominator  and  then  placing  over  it  a  dot. 
Fractional  values  which  could  not  be  expressed  by  any  one  unit- 
fraction  were  expressed  as  the  sum  of  two  or  more  of  them.  Thus,  he 
wrote  I  ^-^  in  place  of  §.  While  Ahmes  knows  |  to  be  equal  to  i  i,  he 
curiously  allows  i  to  appear  often  among  the  unit-fractions  and  adopts 
a  special  symbol  for  it.  The  first  important  problem  naturally  arising 
was,  how  to  represent  any  fractional  value  as  the  sum  of  unit-fractions. 
This  was  solved  by  aid  of  a  table,  given  in  the  papyrus,  in  which  all 

fractions  of  the  form  — 1 —  (where  n  designates  successively  all  the 
2n-\ri 

numbers  up  to  49)  are  reduced  to  the  sum  of  unit-fractions.  Thus, 
2  =^  ^;  -^\  =  T6  T¥¥-  When,  by  whom,  and  how  this  table  was  cal- 
culated, we  do  not  know.  Probably  it  was  compiled  empirically  at 
different  times,  by  different  persons.  It  will  be  seen  that  by  repeated 
appHcation  of  this  table,  a  fraction  whose  numerator  exceeds  two  can 
be  expressed  in  the  desired  form,  provided  that  there  is  a  fraction  in 
the  table  having  the  same  denominator  that  it  has.  Take,  for  ex- 
ample, the  problem,  to  divide  5  by  21.  In  the  first  place,  5  =  1+  2-f  2. 
From  the  table  we  get  2^3- =  Ti^^.  Then^=  2V+(At2)+(A^)  = 
A+  (A  ^)  =  -ir  I  2V  =  T  2 T  =  T  A  T2  •  The  papyrus  contains  prob- 
lems in  which  it  is  required  that  fractions  be  raised  by  addition  or  multi- 
pHcation  to  given  whole  numbers  or  to  other  fractions.  For  example, 
it  is  required  to  increase  ^^J^-^  -^^  to  i.  The  common  denominator 
taken  appears  to  be  45,  for  the  numbers  are  stated  as  iil,  5^  |,  4|-, 
li,  I.  The  sum  of  these  is  23I-  i  i  forty-fifths.  Add  to  this  i  ^,  and 
the  sirni  is  |.  Add  |,  and  we  have  i .  Hence  the  quantity  to  be  added 
to  the  given  fraction  is  4- 1  to • 

Ahmes  gives  the  following  example  involving  an  arithmetical 
progression:  "Divide  100  loaves  among  5  persons;  i-  of  what  the  first 
three  get  is  what  the  last  two  get.  What  is  the  difference?  "  Ahmes 
gives  the  solution:  "Make  the  difference  s^;  23,  ly^,  12,  6i,  i. 
Multiply  by  i|;  38^,  29!-,  20,  io|  |-,  4."    How  did  Ahmes  come  upon 


THE  EGYPTIANS 


13 


5I?  Perhaps  thus:  ^  Let  a  and  — d  be  the  first  term  and  the  differ- 
ence in  the  required  arithmetical  progression.,  then  y[a+(a — d)-\- 
{a — 2d)]={a — 7,d)-\-{a — ^d),  whence  (i=5|(a — ^d),  i.  e.  the  dif- 
ference d  is  5>2  times  the  last  term.  Assuming  the  last  term  i,  he 
gets  his  first  progression.  The  sum  is  60,  but  should  be  100;  hence 
multiply  by  i|,  for  60X  i|=  100.  We  have  here  a  method  of  solution 
which  appears  again  later  among  the  Hindus,  Arabs  and  modern 
Europeans — the  famous  method  of  false  position. 

Ahmes  speaks  of  a  ladder  consisting  of  the  numbers  7,  49,  343, 
2401,  16807.  Adjacent  to  these  powers  of  7  are  the  words  picture, 
cat,  mouse,  barley,  measure.  What  is  the  meaning  of  these  mysterious 
data?  Upon  the  consideration  of  the  problem  given  by  Leonardo  of 
Pisa  in  his  Liber  abaci,  3000  years  later:  "7  old  women  go  to  Rome, 
each  woman  has  7  mules,  each  mule  carries  7  sacks,  etc.",  Moritz 
Cantor  offers  the  following  solution  to  the  Ahmes  riddle:  7  persons 
have  each  7  cats,  each  cat  eats  7  mice,  each  mouse  eats  7  ears  of 
barley,  from  each  ear  7  measures  of  corn  may  grow.  How  many 
persons,  cats,  mice,  ears  of  barley,  and  measures  of  corn,  altogether? 
Ahmes  gives  19607  as  the  sum  of  the  geometric  progression.  Thus, 
the  Ahmes  papyrus  discloses  a  knowledge  of  both  arithmetical  and 
geometrical  progression. 

Ahmes  proceeds  to  the  solution  of  equations  of  one  unknown  quan- 
tity.    The  unknown  quantity  is  called  'hau'  or  heap.     Thus  the 

problem,  "heap,  its  i,  its  whole,  it  makes  19,"  i.  e.  -4-^=19.     In 

8x  X 

this  case,  the  solution  is  as  follows:  — =  19;  -=  2I  i;  x=  i6|  |.    But 

in  other  problems,  the  solutions  are  effected  by  various  other  methods. 
It  thus  appears  that  the  beginnings  of  algebra  are  as  ancient  as  those 
of  geometry. 

That  the  period  of  Ahmes  was  a  flowering  time  for  Egyptian  mathe- 
matics appears  from  the  fact  that  there  exist  other  papyri  (more  re- 
cently discovered)  of  the  same  period.  They  were  found  at  Kahun, 
south  of  the  pyramid  of  Illahun.  These  documents  bear  close  re- 
semblance to  Ahmes.  They  contain,  moreover,  examples  of  quadratic 
equations,  the  earliest  of  which  we  have  a  record.  One  of  them  is:  ^ 
A  given  surface  of,  say,  100  units  of  area,  shall  be  represented  as 
the  sum  of  two  squares,  whose  sides  are  to  each  other  as  i :  1.  In 
modem  symbols,  the  problem  is,  to  find  x  and  y,  such  that  x~-Ary"  = 
100  and  x'.y=i:\.  The  solution  rests  upon  the  method  of  false 
position.  Try  x=i  and  y  =  f,  then  x-+y^  =  ||  and  vTf  =1-  -^^^ 
-\/ioo=  10  and  io-r-|=8.    The  rest  of  the  solution  cannot  be  made 

1  M.  Cantor,  op.  cit.,  Vol.  I,  3.  Aufl.,  1907,  p.  78. 

2  Cantor,  op.  cit.  Vol.  I,  1907,  pp.  95,  96. 


Y.',  A  HISTORY  OF  MATHEMATICS 

out,  but  probably  was  x  =  cSX  i ,  J  =  8X  ^  =  0.  This  solution  leads  to 
the  relation  62+8'^=  lol  The  symbol  P  was  used  to  designate 
square  root. 

In  some  ways  similar  to  the  Ahmes  pap>Tus  is  also  the  Akhmim 
papyrus/  WTitten  over  2000  years  later  at  Akhmim,  a  city  on  the 
Nile  in  Upper  Egyi^t.  It  is  in  Greek  and  is  supposed  to  have  been 
written  at  some  time  between  500  and  800,  a.  d.  It  contains,_  besides 
arithmetical  examples,  a  table  for  finding  "unit-fractions,"  like  that 
of  Ahmes.    Unlike  Ahmes,  it  tells  how  the  table  was  constructed.    The 

rule,  expressed  in  modern  symbols,  is  as  follows:  ^  =  ^-^H — y~- 

q-T      P-T' 
For  s=  2,  this  formula  reproduces  part  of  the  table  in  Ahmes. 

The  principal  defect  of  Eg}T3tian  arithmetic  was  the  lack  of  a 
simple,  comprehensive  symbolism— a  defect  which  not  even  the  Greeks 
were  able  to  remove. 

The  Ahmes  papyrus  and  the  other  papyri  of  the  same  period  repre- 
sent the  m.ost  advanced  attainments  of  the  Egyptians  in  arithmetic 
and  geometry.  It  is  remarkable  that  they  should  have  reached  so 
great  proficiency  in  mathematics  at  so  remote  a  period  of  antiquity. 
But  strange,  indeed,  is  the  fact  that,  during  the  next  two  thousand 
years,  they  should  have  made  no  progress  whatsoever  in  it.  The  con- 
clusion forces  itself  upon  us,  that  they  resemble  the  Chinese  in  the 
stationary  character,  not  only  of  their  government,  but  also  of  their 
learning.'  All  the  knowledge  of  geometry  which  they  possessed  when 
■  Greek  scholars  visited  them,  six  centuries  b.  c,  was  doubtless  known 
to  them  two  thousand  years  earlier,  when  they  built  those  stupendous 
and  gigantic  structures — the  pyramids. 

ij.  Baillet,  "Le  papyrus  mathematique  d'Akhmim,"  Memoires  publi«s  par  les 
membres  de  la  mission  arcMologique  franqaise  au  Caire,  T.  IX,  i""  fascicule,  Paris, 
1892,  pp.  1-88.    See  also  Cantor,  op.  cit.  Vol.  I,  1907,  pp.  67,  504. 


THE  GREEKS 

Greek  Geometry 

About  the  seventh  century  b.  c.  an  active  commercial  intercourse 
sprang  up  between  Greece  and  Egypt.  Naturally  there  arose  an 
interchange  of  ideas  as  well  as  of  merchandise.  Greeks,  thirsting  for 
knowledge,  sought  the  Egyptian  priests  for  instruction.  Thales, 
Pythagoras,  Qinopides,  Plato,  Democritus,  Eudoxus,  all  visited  the 
land  of  the  pyramids.  Egyptian  ideas  were  thus  transplanted  across 
the  sea  and  there  stimulated  Greek  thought,  directed  it  into  new  lines, 
and  gave  to  it  a  basis  to  work  upon.  Greek  culture,  therefore,  is  not 
primitive.  Not  only  in  mathematics,  but  also  in  mythology  and 
art,  Hellas  owes  a  debt  to  older  countries.  To  Egypt  Greece  is  in- 
debted, among  other  things,  for  its  elementary  geometry.  But  this 
does  not  lessen  our  admiration  for  the  Greek  mind.  From  the  mo- 
ment that  Hellenic  philosophers  applied  themselves  to  the  study  of 
Egy-ptian  geometry,  this  science  assumed  a  radically  different  aspect. 
"Whatever  we  Greeks  receive,  we  improve  and  perfect,"  says  Plato. 
The  Egyptians  carried  geometry  no  further  than  was  absolutely  neces- 
sary for  their  practical  wants.  The  Greeks,  on  the  other  hand,  had 
within  them  a  strong  speculative  tendency.  They  felt  a  craving  to 
discover  the  reasons  for  things.  They  found  pleasure  in  the  con- 
templation of  ideal  relations,  and  loved  science  as  science. 

Our  sources  of  information  on  the  history  of  Greek  geom.etry  before 
Euclid  consist  merely  of  scattered  notices  in  ancient  writeis.  The 
early  mathematicians,  Thales  and  Pythagoras,  left  behind  no  written 
records  of  their  discoveries.  A  full  history  of  Greek  geometry  and 
astronomy  during  this  period,  written  by  Eudemus,  a  pupil  of  Aris- 
totle, has  been  lost.  It  was  well  known  to  Proclus,  who,  in  his  com- 
mentaries on  Euclid,  gives  a  brief  account  of  it.  This  abstract  con- 
stitutes our  most  reliable  information.  We  shall  quote  it  frequently 
under  the  name  of  Eiidemian  Summary. 

The  Ionic  School 

To  Thales  (640-546  b.  c),  of  Miletus,  one  of  the  "seven  wise  men," 
and  the  founder  of  the  Ionic  school,  falls  the  honor  of  having  intro- 
duced the  study  of  geometry  into  Greece.  During  middle  life  he 
engaged  in  commercial  pursuits,  which  took  him  to  Egypt.  He  is 
said  to  have  resided  there,  and  to  have  studied  the  physical  sciences 
and  mathematics  with  the  Egyptian  priests.  Plutarch  declares  that 
Thales  soon  excelled  his  masters,  and  amazed  King  Amasis  by  measur- 


i6  A  HISTORY  OF  MATHEMATICS 

ing  the  heights  of  the  pyramids  from  their  shadows.  According  to 
Plutarch,  this  was  done  by  considering  that  the  shadow  cast  by  a 
vertical  staff  of  known  length  bears  the  same  ratio  to  the  shadow  of 
the  pyramid  as  the  height  of  the  staff  bears  to  the  height  of  the  pyra- 
mid. This  solution  presupposes  a  knowledge  of  proportion,  and  the 
Ahmes  papyrus  actually  shows  that  the  rudiments  of  proportion  were 
known  to  the  Egyptians.  According  to  Diogenes  Laertius,  the  pyra- 
mids were  measured  by  Thales  in  a  different  way;  viz.  by  finding  the 
length  of  the  shadow  of  the  pyramid  at  the  moment  when  the  shadow 
of  a  staff  was  equal  to  its  own  length.  Probably  both  methods  were 
used. 

The  Eudemian  Summary  ascribes  to  Thales  the  invention  of  the 
theorems  on  the  equality  of  vertical  angles,  the  equality  of  the  angles 
at  the  base  of  an  isosceles  triangle,  the  bisection  of  a  circle  by  any 
diameter,  and  the  congruence  of  two  triangles  having  a  side  and  the  two 
adjacent  angles  equal  respectively.  The  last  theorem,  combined  (we 
have  reason  to  suspect)  with  the  theorem  on  similar  triangles,  he  appHed 
to  the  measurement  of  the  distances  of  ships  from  the  shore.  Thus 
Thales  wa^  the  first  to  apply  theoretical  geometry  to  practical  uses. 
The  theorem  that  all  angles  inscribed  in  a  semicircle  are  right  angles 
is  attributed  by  some  ancient  writers  to  Thales,  by  others  to  Pythag- 
oras. Thales  was  doubtless  familiar  with  other  theorems,  not  re- 
corded by  the  ancients.  It  has  been  inferred  that  he  knew  the  sum 
of  the  three  angles  of  a  triangle  to  be  equal  to  two  right  angles,  and 
the  sides  of  equiangular  triangles  to  be  proportional.^  The  Egyptians 
must  have  made  use  of  the  above  theorems  on  the  straight  line,  in 
some  of  their  constructions  found  in  the  Ahmes  papyrus,  but  it  was 
left  for  the  Greek  philosopher  to  give  these  truths,  which  others  saw, 
but  did  not  formulate  into  words,  an  explicit,  abstract  expression,  and 
to  put  into  scientific  language  and  subject  to  proof  that  which  others 
merely  felt  to  be  true.  Thales  may  be  said  to  have  created  the  geom- 
etry of  lines,  essentially  abstract  in  its  character,  while  the  Egyptians 
studied  only  the  geometry  of  surfaces  and  the  rudiments  of  solid 
geometry,  empirical  in  their  character.  2 

With  Thales  begins  also  the  study  of  scientific  astronomy.  He 
acquired  great  celebrity  by  the  prediction  of  a  solar  eclipse  in  585  b.  c. 
Whether  he  predicted  the  day  of  the  occurrence,  or  simply  the  year, 
is  not  known.  It  is  told  of  him  that  while  contemplating  the  stars 
during  an  evening  walk,  he  fell  into  a  ditch.  The  good  old  woman 
attending  him  exclaimed,  ''How  canst  thou  know  what  is  doing  in 
the  heavens,  when  thou  seest  not  what  is  at  thy  feet?  " 

The  two  most  prominent  pupils  of  Thales  were  Anaximander  (b.  611 

1  G.  J.  Allman,  Greek  Geometry  from  Thales  to  Euclid.  Dublin,  1889,  p.  10. 
George  Johnston  Allman  (1824-1904)  was  professor  of  mathematics  at  Queen's 
College,  Galway,  Ireland. 

2  G.  J.  Allman,  op.  cit.,  p.  15. 


GREEK  GEOMETRY  17 

B.  c.)  and  Anaximenes  (b.  570  b.  c).  They  studied  chiefly  astronomy 
and  physical  philosophy.  Of  Anaxagoras  (500-428  b.  c),  a  pupil  of 
Anaximenes,  and  the  last  philosopher  of  the  Ionic  school,  we  know 
Uttle,  except  that,  while  in  prison,  he  passed  his  time  attempting  to 
square  the  circle.  This  is  the  first  time,  in  the  history  of  mathematics, 
that  we  find  mention  of  the  famous  problem  of  the  quadrature  of  the 
circle,  that  rock  upon  which  so  many  reputations  have  been  destroyed. 
It  turns  upon  the  determination  of  the  exact  value  of  tt.  Approxima- 
tions to  TT  had  been  made  by  the  Chinese,  Babylonians,  Hebrews,  and 
Egyptians.  But  the  invention  of  a  method  to  find  its  exact  value,  is 
the  knotty  problem  which  has  engaged  the  attention  of  many  minds 
from  the  time  of  Anaxagoras  down  to  our  own.  Anaxagoras  did 
not  offer  any  solution  of  it,  and  seems  to  have  luckily  escaped  par- 
alogisms. The  problem  soon  attracted  popular  attention,  as  appears 
from  the  reference  to  it  made  in  414  B.  c.  by  the  comic  poet  Aris- 
tophanes in  his  play,  the  "Birds."  ^ 

About  the  time  of  Anaxagoras,  but  isolated  from  the  Ionic  school, 
flourished  CEnopides  of  Chios.  Proclus  ascribes  to  him  the  solution 
of  the  following  problems:  From  a  point  without,  to  draw  a  per- 
pendicular to  a  given  line,  and  to  draw  an  angle  on  a  line  equal  to  a 
given  angle.  That  a  man  could  gain  a  reputation  by  solving  problems 
so  elementary  as  these,  indicates  that  geometry  was  still  in  its  infancy, 
and  that  the  Greeks  had  not  yet  gotten  far  beyond  the  Egyptian  con- 
structions. 

The  Ionic  school  lasted  over  one  hundred  years.  The  progress  of 
mathematics  during  that  period  was  slow,  as  compared  "v^ith  its 
growth  in  a  later  epoch  of  Greek  history.  A  new  impetus  to  its  prog- 
ress was  given  by  Pythagoras. 

The  School  of  Pythagoras 

Pythagoras  (58o?-5oo?  b.  c.)  was  one  of  those  figures  which  im- 
pressed the  imagination  of  succeeding  times  to  such  an  extent  that 
their  real  histories  have  become  difficult  to  be  discerned  through  the 
mythical  haze  that  envelops  them.  The  following  account  of  Pythag- 
oras excludes  the  most  doubtful  statements.  He  was  a  native  of 
Samos,  and  was  drawn  by  the  fame  of  Pherecydes  to  the  island  of 
Syros.  He  then  visited  the  ancient  Thales,  who  incited  him  to  study 
in  Egypt.  He  sojourned  in  Egypt  many  years,  and  may  have  visited 
Babylon.  On  his  return  to  Samos,  he  found  it  under  the  tyranny  of 
Polycrates.  Failing  in  an  attempt  to  found  a  school  there,  he  quitted 
home  again  and,  following  the  current  of  civilisation,  removed  to 
Magna  Grascia  in  South  Italy.  He  settled  at  Croton,  and  founded 
the  famous  Pythagorean  school.    This  was  not  merely  an  academy  for 

1  F.  Rudio  in  Bibliolhcca  malhcmalica,  3  S.,  Vol.  8,  1907-S,  pp.  13-22. 


i8  A  HISTORY  OF  MATHEMATICS 

the  teaching  of  philosophy,  mathematics,  and  natural  science,  but  it 
was  a  brotherhood,  the  members  of  which  were  united  for  life.  This 
brotherhood  had  observances  approaching  masonic  peculiarity.  They 
were  forbidden  to  divulge  the  discoveries  and  doctrines  of  their  school. 
Hence  we  are  obliged  to  speak  of  the  Pythagoreans  as  a  body,  and 
find  it  difficult  to  determine  to  whom  each  particular  discovery  is  to 
be  ascribed.  The  Pythagoreans  themselves  were  in  the  habit  of  re- 
ferring every  discovery  back  to  the  great  founder  of  the  sect. 

This  school  grew  rapidly  and  gained  considerable  political  ascend- 
ency. But  the  mystic  and  secret  observances,  introduced  in  imitation 
of  Egyptian  usages,  and  the  aristocratic  tendencies  of  the  school, 
caused  it  to  become  an  object  of  suspicion.  The  democratic  party  in 
Lower  Italy  revolted  and  destroyed  the  buildings  of  the  Pythagorean 
school.  Pythagoras  fled  to  Tarentum  and  thence  to  Metapontum, 
where  he  was  murdered. 

Pythagoras  has  left  behind  no  mathematical  treatises,  and  our 
sources  of  information  are  rather  scanty.  Certain  it  is  that,  in  the 
Pythagorean  school,  mathematics  was  the  principal  study.  Pythag- 
oras raised  mathematics  to  the  rank  of  a  science.  Arithmetic  was 
courted  by  him  as  fervently  as  geometry.  In  fact,  arithmetic  is  the 
foundation  of  his  philosophic  system. 

The  Eudemian  Summary  says  that  "Pythagoras  changed  the  study 
of  geometry  into  the  form  of  a  liberal  education,  for  he  examined  its 
principles  to  the  bottom,  and  investigated  its  theorems  in  an  imma- 
terial and  intellectual  manner."  His  geometry  was  connected  closely 
with  his  arithmetic.  He  was  especially  fond  of  those  geometrical 
relations  which  admitted  of  arithmetical  expression. 

Like  Egyptian  geometry,  the  geometry  of  the  Pythagoreans  is  much 
concerned  with  areas.  To  Pythagoras  is  ascribed  the  important 
theorem  that  the  square  on  the  hypotenuse  of  a  right  triangle  is 
equal  to  the  sum  of  the  squares  on  the  other  two  sides.  He  had 
probably  learned  from  the  Egyptians  the  truth  of  the  theorem  in  the 
special  case  when  the  sides  are  3,  4,  5,  respectively.  The  story  goes, 
that  Pythagoras  was  so  jubilant  over  this  discovery  that  he  sacrificed 
a  hecatomb.  Its  authenticity  is  doubted,  because  the  Pythagoreans 
believed  in  the  transmigration  of  the  soul  and  opposed  the  shedding 
of  blood.  In  the  later  traditions  of  the  Neo-Pythagoreans  this  ob- 
jection is  removed  by  replacing  this  bloody  sacrifice  by  that  of  "an 
ox  made  of  flour!"  The  proof  of  the  law  of  three  squares,  given  in 
Euclid's  Elements,  I.  47,  is  due  to  Euclid  himself,  and  not  to  the 
Pythagoreans.  What  the  Pythagorean  method  of  proof  was  has 
been  a  favorite  topic  for  conjecture. 

The  theorem  on  the  sum  of  the  three  angles  of  a  triangle,  presum- 
ably known  to  Thales,  was  proved  by  the  Pythagoreans  after  the 
manner  of  Euclid.  They  demonstrated  also  that  the  plane  about  a 
point  is  completely  filled  by  six  equilateral  triangles,  four  squares,  or 


GREEK  GEOMETRY  19 

chree  regular  hexagons,  so  that  it  is  possible  to  divide  up  a  plane  into 
figures  of  either  kind. 

From  the  equilateral  triangle  and  the  square  arise  the  solids,  namely, 
the  tetraedron,  octaedron,  icosaedron,  and  the  cube.  These  solids 
were,  in  all  probability,  known  to  the  Egyptians,  excepting,  perhaps, 
the  icosaedron.  In  Pythagorean  philosophy,  they  represent  respec- 
tively the  four  elements  of  the  physical  world ;  namely,  fire,  air,  water, 
and  earth.  Later  another  regular  solid  was  discovered,  namely,  the 
dodecaedron,  which,  in  absence  of  a  fifth  element,  was  made  to  repre- 
sent the  universe  itself.  lamblichus  states  that  Hippasus,  a  Pytha- 
gorean, perished  in  the  sea,  because  he  boasted  that  he  first  di\ailged 
"the  sphere  with  the  twelve  pentagons."  The  same  story  of  death  at 
sea  is  told  of  a  Pythagorean  who  disclosed  the  theory  of  irrationals. 
The  star-shaped  pentagram  was  used  as  a  symbol  of  recognition  by 
the  Pythagoreans,  and  was  called  by  them  Health. 

Pythagoras  called  the  sphere  the  most  beautiful  of  all  solids,  and 
the  circle  the  most  beautiful  of  all  plane  figures.  The  treatment  of 
the  subjects  of  proportion  and  of  irrational  quantities  by  him  and 
his  school  will  be  taken  up  under  the  head  of  arithmetic. 

According  to  Eudemus,  the  Pythagoreans  invented  the  problems 
concerning  the  application  of  areas,  including  the  cases  of  defect  and 
excess,  as  in  Euclid,  VI.  28,  29. 

They  were  also  familiar  with  the  construction  of  a  polygon  equal 
in  area  to  a  given  polygon  and  similar  to  another  given  polygon.  This 
problem  depends  upon  several  important  and  somewhat  advanced 
theorems,  and  testifies  to  the  fact  that  the  Pythagoreans  made  no 
mean  progress  in  geometry. 

Of  the  theorems  generally  ascribed  to  the  Italian  school,  some 
cannot  be  attributed  to  Pythagoras  himself,  nor  to  his  earliest  suc- 
cessors. The  progress  from  empirical  to  reasoned  solutions  must,  of 
necessity,  have  been  slow.  It  is  worth  noticing  that  on  the  circle 
no  theorem  of  any  importance  was  discovered  by  this  school. 

Though  politics  broke  up  the  Pythagorean  fraternity,  yet  the  school 
continued  to  exist  at  least  two  centuries  longer.  Among  the  later 
Pythagoreans,  Philolaus  and  Archytas  are  the  most  prominent. 
Philolaus  wTote  a  book  on  the  Pythagorean  doctrines.  By  him  were 
first  given  to  the  world  the  teachings  of  the  Italian  school,  which  had 
been  kept  secret  for  a  whole  century.  The  brilliant  Archjrtas  (42cS- 
347  B.  c.)  of  Tarentum,  known  as  a  great  statesman  and  general,  and 
universally  admired  for  his  virtues,  was  the  only  great  geometer 
among  the  Greeks  when  Plato  opened  his  school.  Archytas  was  the 
first  to  apply  geometry  to  mechanics  and  to  treat  the  latter  subject 
methodically.  He  also  found  a  very  ingenious  mechanical  solution 
to  the  problem  of  the  duplication  of  the  cube.  His  solution  involves 
clear  notions  on  the  generation  of  cones  and  cylinders.  This  problem 
reduces  itself  to  finding  two  mean  proportionals  between  two  given 


2c  A  HISTORY  OF  MATHEMATICS 

lines.  These  mean  proportionals  were  obtained  by  Archytas  from 
the  section  of  a  half-cylinder.  The  doctrine  of  proportion  was  ad- 
vanced through  him. 

There  is  every  reason  to  believe  that  the  later  Pythagoreans  exer- 
cised a  strong  influence  on  the  study  and  development  of  mathematics 
at  Athens.  The  Sophists  acquired  geometry  from  Pythagorean 
sources.  Plato  bought  the  works  of  Philolaus,  and  had  a  warm  friend 
in  Archytas. 

The  Sophist  School 

After  the  defeat  of  the  Persians  under  Xerxes  at  the  battle  of 
Salamis,  480  b.  c,  a  league  was  formed  among  the  Greeks  to  preserve 
the  freedom  of  the  now  liberated  Greek  cities  on  the  islands  and  coast 
of  the  yEgagan  Sea.  Of  this  league  Athens  soon  became  leader  and 
dictator.  She  caused  the  separate  treasury  of  the  league  to  be  merged 
into  that  of  Athens,  and  then  spent  the  money  of  her  allies  for  her 
own  aggrandisement.  Athens  was  also  a  great  commercial  centre. 
Thus  she  became  the  richest  and  most  beautiful  city  of  antiquity. 
All  menial  w^ork  was  performed  by  slaves.  The  citizen  of  Athens  was 
well-to-do  and  enjoyed  a  large  amount  of  leisure.  The  government 
being  purely  democratic,  every  citizen  was  a  politician.  To  make  his 
influence  felt  among  his  fellow-men  he  must,  first  of  all,  be  educated. 
Thus  there  arose  a  demand  for  teachers.  The  supply  came  principally 
from  Sicily,  where  Pythagorean  doctrines  had  spread.  These  teachers 
w^ere  called  Sophists,  or  "wise  men."  Unlike  the  Pythagoreans,  they 
accepted  pay  for  their  teaching.  Although  rhetoric  was  the  principal 
feature  of  their  instruction,  they  also  taught  geometr}^  astronomy, 
and  philosophy.  Athens  soon  became  the  headquarters  of  Grecian 
men  of  letters,  and  of  mathematicians  in  particular.  The  home  of 
mathematics  among  the  Greeks  was  first  in  the  Ionian  Islands,  then 
in  Lower  Italy,  and  during  the  time  now  under  consideration,  at 
Athens. 

The  geometry  of  the  circle,  which  had  been  entirely  neglected  by 
the  Pythagoreans,  was  taken  up  by  the  Sophists.  Nearly  all  their 
discoveries  were  made  in  connection  with  their  innumerable  attempts 
to  solve  the  following  three  famous  problems: — 

(i)  To  trisect  an  arc  or  an  angle. 

(2)  To  "double  the  cube,"  i.  e.,  to  find  a  cube  whose  vohime  is 
double  that  of  a  given  cube. 

(3)  To  "square  the  circle,"  i.  e.  to  find  a  square  or  some  other 
rectilinear  figure  exactly  equal  in  area  to  a  given  circle. 

These  problems  have  probably  been  the  subject  of  more  discussion 
and  research  than  any  other  problems  in  mathematics.  The  bisection 
of  an  angle  was  one  of  the  easiest  problems  in  geometry.  The  trisec- 
ticn  of  an  angle,  on  the  other  hand,  presented  unexpected  difficulties. 
A  right  angle  had  been  divided  into  three  equal  parts  by  the  Pytha- 


GREEK   GEOMETRY  21 

gorcans.  But  the  general  construction,  though  easy  in  appearance, 
cannot  be  effected  by  the  aid  onl)'  of  ruler  and  compasses.  Among 
the  first  to  wrestle  with  it  was  Hippias  of  Elis,  a  contemporary  of 
Socrates,  and  born  about  460  b.  c.  Unable  to  reach  a  solution  by 
ruler  and  compasses  only,  he  and  o'.her  Greek  geometers  r^sort.d  to 
the  use  of  other  means.  Proclus  mentions  a  man,  liippias,  presum- 
ably Ilippias  of  Elis,  as  the  inventor  of  a  transcend^^ntai  curve  which 
served  to  divide  an  an;":le  not  only  into  llircc,  but  into  any  number  of 
equal  parts.  This  se.me  curve  v/as  used  later  b>'  Dinostratus  and 
others  for  the  quadniture  of  the  circle.  On  this  account  it  is  called 
the  quadratrlx.  The  curve  may  be  described  thus:  The  side  AB  of  the 
square  shown  in  the  figure  turns  uniformly  about  A,  the  point  B 
moving  e.long  the  circular  arc  BED.  In  the 
same  tir.ie,  the  side  BC  moves  parallel  to  it- 
self and  uniformly  from  the  position  of  BC 
to  that  of  AD.  The  locus  of  intersection  of 
AB  and  BC,  when  thus  moving,  is  the 
quadratrix  BFG.    Its  equation  we  now  write 

y  =  .vcot — '- .    The  ancients  considered  only 

the  part  of  the  curve  that  lies  inside  the 

quadrant  of  the  circle;  they  did  not  know 

that  X  =  =t:  2r  are  asymptotes,  nor  that  there 

is  an  infinite  number  of  branches.    According  to  Pappus,  Dinostratus 

effected  the  quadrature  by  establishing  the  theorem  that  BED:  AD 

=  AD:AG. 

The  Pythagoreans  had  shown  that  the  diagonal  of  a  square  is  the 
side  of  another  square  having  double  the  area  of  the  original  one. 
This  probably  suggested  the  problem  of  the  duplication  of  the  cube, 
i.  c,  to  find  the  edge  of  a  cube  having  double  the  volume  of  a  given 
cube.  Eratosthenes  ascribes  to  this  problem  a  different  origin.  The 
Delians  were  once  suffering  from  a  pestilence  and  were  ordered  by 
the  oracle  to  double  a  certain  cubical  altar.  Thoughtless  worknien 
simply  constructed  a  cube  with  edges  twice  as  long,  but  brainless 
work  like  that  did  not  pacify  the  gods.  The  error  being  discovered, 
Plato  was  consulted  on  the  matter.  He  and  his  disciples  searched 
eagerly  for  a  solution  to  this  "Delian  Problem."  An  important  con- 
tribution to  this  problem  was  made  by  Hippocrates  of  Chios  (about 
430  B.  c).  He  was  a  talented  mathematician  but,  having  been  de- 
frauded, of  his  property,  he  was  pronounced  slow  and  stupid.  It  is 
also  s4d  of  him  that  he  was  the  first  to  accept  pay  for  the  teaching  of 
mathematics.  He  showed  that  the  Delian  Problem  could  be  reduced 
to  finding  two  mean  proportionals  between  a  given  line  and  another 
twice  as  long.  For,  in  the  proportion  a  •.x  =  x  •.y=y  :  2a,  since  x-  = 
ay  and  y'^=2ax  and  x^=a-y',  we  have  x'^=2a^x  and  x^  =  2a^.     But, 


22  A  HISTORY  OF  MATHEMATICS 

of  course,  he  failed  to  find  the  two  mean  proportionals  by  geometric 
construction  with  ruLr  ii.sd  compasses.  He  made  himself  celebrated 
by  squaring  certain  lunes.  According  to  Simplicius,  Hippocrates 
believed  he  actual b'  succeeded  in  aj^plying  one  of  his  lune-quadratures 
to  the  quadrature  of  the  circle.  That  Hippocrates  really  committed 
this  fallacy  is  not  generally  accepted. 

In  the  first  lune  which  he  squared,  he  took  an  isosceles  triangle 
ABC,  right-angled  at  C,  and  drew  a  semi-circle  on  AB  as  a  diameter, 
and  passing  through  C.  He  drew  also  a  semi-circle  on  AC  as  a  diam- 
eter and  lying  outside  the  triangle  ABC.  The  lunar  area  thus  formed 
is  half  the  area  of  the  triangle  ABC.  This  is  the  first  example  of  a 
curvilinear  area  which  admits  of  exact  quadrature.  Hippocrates 
squared  other  lunes,  hoping,  no  doubt,  that  he  might  be  led  to  the 
quadrature  of  the  circle.^  In  1840  Th.  Clausen  found  other  quadrable 
lunes,  but  in  1902  E.  Landau  of  Gottingen  pointed  out  that  two  of 
the  four  lunes  which  Clausen  supposed  to  be  new,  were  known  to 
Hippocrates.^ 

In  his  study  of  the  quadrature  and  duplication-problems,  Hip- 
pocrates contributed  much  to  the  geometry  of  the  circle.  He  showed 
that  circles  are  to  each  other  as  the  squares  of  their  diameters,  that 
similar  segments  in  a  circle  are  as  the  squares  of  their  chords  and 
contain  equal  angles,  that  in  a  segment  less  than  a  semi-circle  the 
angle  is  obtuse.  Hippocrates  contributed  vastly  to  the  logic  of  geom- 
etry. His  investigations  are  the  oldest  "reasoned  geometrical  proofs 
in  existence  "  (Gow) .  For  the  purpose  of  describing  geometrical  figures 
he  used  letters,  a  practice  probably  introduced  by  the   Pythagoreans. 

The  subject  of  similar  figures,  as  developed  by  Hippocrates,  in- 
volved the  theory  of  proportion.  Proportion  had,  thus  far,  been  used 
by  the  Greeks  only  in  numbers.  They  never  succeeded  in  uniting 
the  notions  of  numbers  and  magnitudes.  The  term  "number"  was 
used  by  them  in  a  restricted  sense.  What  we  call  irrational  numbers 
was  not  included  under  this  notion.  Not  even  rational  fractions 
were  called  numbers.  They  used  the  word  in  the  same  sense  as  we 
use  "positive  integers."  Hence  numbers  were  conceived  as  discon- 
tinuous, while  magnitudes  were  continuous.  The  two  notions  ap- 
peared, therefore,  entirely  distinct.  The  chasm  between  them  is  ex- 
posed to  full  view  in  the  statement  of  Euclid  that  "incommensurable 
magnitudes  do  not  have  the  same  ratio  as  numbers."  In  Euclid's 
Elements  we  find  the  theory  of  proportion  of  magnitudes  developed 
and  treated  independent  of  that  of  numbers.  The  transfer  of  the 
theory  of  pro])ortion  from  numbers  to  magnitudes  (and  to  lengths  in 
particular)  was  a  difficult  and  important  step. 

'  A  full  account  is  given  by  G.  Loria  in  his  Le  scicnze  esalle  nelVantica  Grecia, 
Milano,  2  edition,  1914,  pp.  74-94.  Loria  gives  also  full  bibliographical  references 
to  the  extensi\'e  literature  on  Iiip[)ocrates. 

^  E.  W.  Hobson,  Squaring  the  Circle,  Cambridge,  19 13,  p.  16. 


GREEK  GEOMETRY  23 

Hippocrates  added  to  his  fame  by  writing  a  geometrical  text-book, 
called  the  Elements.  This  publication  shows  that  the  Pythagorean 
habit  of  secrecy  was  being  abandoned;  secrecy  was  contrary  to  the 
spirit  of  Athenian  life. 

The  sophist  Antiphon,  a  contemporary  of  Hippocrates,  introduced 
the  process  of  exhaustion  for  the  purpose  of  solving  the  problem  of 
the  quadrature.  He  did  himself  credit  by  remarking  that  by  inscrib- 
ing in  a  circle  a  square  or  an  equilateral  triangle,  and  on  its  sides 
erecting  isosceles  triangles  with  their  vertices  in  the  circumference, 
and  on  the  sides  of  these  triangles  erecting  new  triangles,  etc.,  one 
could  obtain  a  succession  of  regular  polygons,  of  which  each  approaches 
nearer  to  the  area  of  the  circle  than  the  previous  one,  until  the  circle 
is  finally  exhausted.  Thus  is  obtained  an  inscribed  polygon  whose 
sides  coincide  with  the  circumference.  Since  there  can  be  found 
squares  equal  in  area  to  any  polygon,  there  also  can  be  found  a  square 
equal  to  the  last  polygon  inscribed,  and  therefore  equal  to  the  circle 
itself,  Bryson  of  Heraclea,  a  contemporary  of  Antiphon,  advanced 
the  problem  of  the  quadrature  considerably  by  circumscribing  poly- 
gons at  the  same  time  that  he  inscribed  polygons.  He  erred,  however, 
in  assuming  that  the  area  of  a  circle  was  the  arithmetical  mean  be- 
tween circumscribed  and  inscribed  polygons.  Unlike  Brj'son  and 
the  rest  of  Greek  geometers,  Antiphon  seems  to  have  believed  it 
possible,  by  continually  doubling  the  sides  of  an  inscribed  polygon, 
to  obtain  a  polygon  coinciding  with  the  circle.  This  question  gave 
rise  to  lively  disputes  in  Athens.  If  a  polygon  can  coincide  with  the 
circle,  then,  says  Simplicius,  we  must  put  aside  the  notion  that  magni- 
tudes are  divisible  ad  injhiitiim.  This  difficult  philosophical  question 
led  to  paradoxies  that  are  difficult  to  explain  and  that  deterred  Greek 
mathematicians  from  introducing  ideas  of  infinity  into  their  geometry; 
rigor  in  geometric  proofs  demanded  the  exclusion  of  obscure  concep- 
tions. Famous  are  the  arguments  against  the  possibility  of  motion 
that  were  advanced  by  Zeno  of  Elea,  the  great  dialectician  (early  in 
the  5th  century  B.  c).  None  of  Zeno's  writings  have  come  down  to 
us.  We  know  of  his  tenets  only  through  his  critics,  Plato,  Aristotle, 
Simplicius.  Aristotle,  in  his  Physics,  VI,  9,  ascribes  to  Zeno  four 
arguments,  called  "Zeno's  paradoxies":  (i)  The  "Dichotomy":  You 
cannot  traverse  an  infinite  number  of  points  in  a  finite  time;  you 
must  traverse  the  half  of  a  given  distance  before  you  traverse  the 
whole,  and  the  half  of  that  again  before  you  can  traverse  the  whole. 
This  goes  on  ad  infinitum,  so  that  (if  space  is  made  up  of  points)  there 
is  an  infinite  number  in  any  given  space,  and  it  cannot  be  traversed 
in  a  finite  time.  (2)  The  "Achilles":  Achilles  cannot  overtake  a  tor- 
toise. For,  Achilles  must  first  reach  the  place  from  which  the  tortoise 
started.  By  that  time  the  tortoise  will  have  moved  on  a  little  way. 
Achilles  must  then  traverse  that,  and  still  the  tortoise  will  be  ahead. 
He  is  always  nearer,  yet  never  makes  up  to  it.     (3)  The  "Arrow": 


24  A  HISTORY  OF  MATHEMATICS 

An  arrow  in  any  given  moment  of  its  flight  must  be  at  rest  in  some 
particular  point.  (4)  The  "Stade":  Suppose  three  parallel  rows  of 
points  in  juxtaposition,  as  in  Fig.  i.    One  of  these  (B)  is  immovable, 

A     .     .     .     .  ^A     .     .     .     . 

B     .     .     .     .  B  .... 

C     .     .     .     .  C  .     .     .     .-^ 

Fig.  I  Fig.  2 

while  A  and  C  move  in  opposite  directions  with  equal  velocity,  so 
as  to  come  into  the  position  in  Fig.  2.  The  movement  of  C  relatively 
to  A  will  be  double  its  movement  relatively  to  B,  or,  in  other  words, 
any  given  point  in  C  has  passed  twice  as  many  points  in  A  as  it  has 
in  B.  It  cannot,  therefore,  be  the  case  that  an  instant  of  time  corre- 
sponds to  the  passage  from  one  point  to  another. 

Plato  says  that  Zeno's  purpose  was  "  to  protect  the  arguments  of 
Parmenides  against  those  who  make  fun  of  him";  Zeno  argues  that 
"  there  is  no  many,''  he  "  denies  plurality."  That  Zeno's  reasoning  was 
wrong  has  been  the  view  universally  held  since  the  time  of  Aristotle 
down  to  the  middle  of  the  nineteenth  century.  More  recently  the 
opinion  has  been  advanced  that  Zeno  was  incompletely  and  incor- 
rectly reported,  that  his  arguments  are  serious  efforts,  conducted  with 
logical  rigor.  This  view  has  been  advanced  by  Cousin,  Grote  and  P. 
Tannery.^  Tannery  claims  that  Zeno  did  not  deny  motion,  but 
wanted  to  show  that  motion  was  impossible  under  the  Pythagorean 
conception  of  space  as  the  sum  of  points,  that  the  four  arguments  must 
be  taken  together  as  constituting  a  dialogue  between  Zeno  and- an 
adversary  and  that  the  arguments  are  in  the  form  of  a  double  dilemma 
into  which  Zeno  forces  his  adversary.  Zeno's  arguments  involve  con- 
cepts of  continuity,  of  the  infinite  and  infinitesimal;  they  are  as  much 
the  subjects  of  debate  now  as  they  were  in  the  time  of  Aristotle. 
Aristotle  did  not  successfully  explain  Zeno's  paradoxes.  He  gave  no 
reply  to  the  query  arising  in  the  mind  of  the  student,  how  is  it  pos- 
sible for  a  variable  to  reach  its  Hmit?  Aristotle's  continuum  was  a 
sensuous,  physical  one;  he  held  that,  since  a  line  cannot  be  built  up 
of  points,  a  line  cannot  actually  be  subdivided  into  points.  "The 
continued  bisection  of  a  quantity  is  unlimited,  so  that  the  unlimited 
exists  potentially,  but  is  actually  never  reached."  No  satisfactory 
explanation  of  Zeno's  arguments  was  given  before  the  creation  of 
Georg  Cantor's  continuum  and  theory  of  aggregates. 

The  process  of  exhaustion  due  to  Antiphon  and  Bryson  gave  rise 
to  the  cumbrous  but  perfectly  rigorous  "method  of  exhaustion."  In 
determining  the  ratio  of  the  areas  between  two  curvilinear  plane 
figures,  say  two  circles,  geometers  first  inscribed  or  circumscribed 
similar  polygons,  and  then  by  increasing  indefinitely  the  number  of 

1  See  F.  Cajori,  "The  History  of  Zeno's  Arguments  on  Motion"  in  the  Americ. 
Malh.  Monthly,  Vol.  22,  1915,  p.  3. 


GREEK  GEOMETRY  25 

sides,  nearly  exhausted  the  spaces  between  the  polygons  and  circum- 
ferences. From  the  theorem  that  similar  polygons  inscribed  in  circles 
are  to  each  other  as  the  squares  on  their  diameters,  geometers  may 
have  divined  the  theorem  attributed  to  Hippocrates  of  Chios  that  the 
circles,  which  differ  but  little  from  the  last  drawn  polygons,  must  be 
to  each  other  as  the  squares  on  their  diameters.  But  in  order  to  ex- 
clude all  vagueness  and  possibility  of  doubt,  later  Greek  geometers 
applied  reasoning  like  that  in  Euclid,  XII,  2,  as  follows:  Let  C  and  c, 
D  and  d  be  respectively  the  circles  and  diameters  in  question.  Then 
if  the  proportion  D'^  •.d-  =  C  :  c  is  not  true,  suppose  that  D"^  •.d'  =  C  :  c^. 
If  c^<c,  then  a  polygon  p  can  be  inscribed  in  the  circle  c  which  comes 
nearer  to  it  in  area  than  does  c^.  If  P  be  the  corresponding  polygon 
in  C,  then  P  :p  =  D^-:  d-  =  C  :  c\  and  P  :C=p  :  c^.  Since  p>cK  we 
have  P>  C,  which  is  absurd.  Next  they  proved  by  this  same  method  of 
^■ediictio  ad  absnrdum  the  falsity  of  the  supposition  that  c^>c.  Since 
c^  can  be  neither  larger  nor  smaller  than  c,  it  must  be  equal  to  it, 
Q.E.D.  Hankel  refers  this  IMethod  of  Exhaustion  back  to  Hippocrates 
of  Chios,  but  the  reasons  for  assigning  it  to  this  early  WTiter,  rather 
than  to  Eudoxus,  seem  insufficient. 

Though  progress  in  geometry  at  this  period  is  traceable  only  at 
Athens,  yet  Ionia,  Sicily,  Abdera  in  Thrace,  and  Cyrene  produced 
mathematicians  who  made  creditable  contributions  to  the  science.  We 
can  mention  here  only  Democritus  of  Abdera  (about  460-370  b.  c), 
a  pupil  of  Anaxagoras,  a  friend  of  Philolaus,  and  an  admirer  of  the 
Pythagoreans.  He  visited  Egj'pt  and  perhaps  even  Persia.  He  was 
a  successful  geometer  and  wrote  on  incommensurable  lines,  on  geom- 
etry, on  numbers,  and  on  perspective.  None  of  these  works  are  extant. 
Pie  used  to  boast  that  in  the  construction  of  plane  figures  with  proof 
no  one  had  yet  surpassed  him,  not  even  the  so-called  harpedonaptae 
("rope-stretchers")  of  Egypt.  By  this  assertion  he  pays  a  flattering 
compliment  to  the  skill  and  ability  of  the  Egyptians. 

The  Platonic  School 

During  the  Peloponnesian  War  (431-404  b.  c.)  the  progress  of  geom- 
etry was  checked.  After  the  war,  Athens  sank  into  the  background 
as  a  minor  political  power,  but  advanced  more  and  more  to  the  front 
as  the  leader  in  philosophy,  literature,  and  science.  Plato  was  born 
at  Athens  in  429  b.  c,  the  year  of  the  great  plague,  and  died  in  348. 
He  was  a  pupil  and  near  friend  of  Socrates,  but  it  was  not  from  him 
that  he  acquired  his  taste  for  mathematics.  After  the  death  of 
Socrates,  Plato  travelled  extensively.  In  Cyrene  he  studied  mathe- 
matics under  Thcodorus.  He  went  to  Egy]:)t,  then  to  Lower  Italy 
and  Sicily,  where  he  came  in  contact  with  the  Pythagoreans.^  Archytas 
of  Tarentum  and  Timceus  of  Locri  became  his  i:Uimate  friends.  On 
nis  return  to  Athens,  about  389  b.  c,  he  founded  his  school  in  the 


26  A  HISTORY  OF  MATHEMATICS 

groves  of  the  Academia,  and  devoted  the  remainder  of  his  Hfe  to  teach- 
ing and  writing. 

•  Plato's  physical  philosophy  is  partly  based  on  that  of  the  Pytha- 
goreans. Like  them,  he  sought  in  arithmetic  and  geometry  the  key 
to  the  universe.*  When  questioned  about  the  occupation  of  the  Deity, 
Plato  answered  that  "He  geometrises  continually."  Accordingly,  a 
knowledge  of  geometry  is  a  necessary  preparation  for  the  study  of 
philosophy.  To  show  how  great  a  value  he  put  on  mathematics  and 
how  necessary  it  is  for  higher  speculation,  Plato  placed  the  inscrip- 
tion over  his  porch,  "Let  no  one  who  is  unacquainted  with  geometry 
enter  here."  Xenocrates,  a  successor  of  Plato  as  teacher  in  the 
Academy,  followed  in  his  master's  footsteps,  by  declining  to  admit  a 
pupil  who  had  no  mathematical  training,  with  the  remark,  "Depart, 
for  thou  hast  not  the  grip  of  philosophy."  Plato  observed  that  geom- 
etry trained  the  mind  for  correct  and  vigorous  thinking.  Hence  it 
was  that  the  Eudemian  Summary  says,  "He  filled  his  Avritings  with 
mathematical  discoveries,  and  exhibited  on  every  occasion  the  re- 
markable connection  between  mathematics  and  philosophy." 

With  Plato  as  the  head-master,  we  need  not  wonder  that  the  Pla- 
tonic school  produced  so  large  a  number  of  mathematicians.  Plato 
did  little  real  original  work,  but  he  made  valuable  improvements  in 
the  logic  and  methods  employed  in  geometry.  It  is  true  that  the 
Sophist  geometers  of  the  previous  century  were  fairly  rigorous  in  their 
proofs,  but  as  a  rule  they  did  not  reflect  on  the  inward  nature  of  their 
methods.  They  used  the  axioms  without  giving  them  explicit  ex- 
pression, and  the  geometrical  concepts,  such  as  the  point,  line,  surface, 
etc.,  without  assigning  to  them  formal  definitions. ^  The  Pythagoreans 
called  a  point  "unity  in  position,"  but  this  is  a  statement  of  a  philo- 
sophical theory  rather  than  a  definition.  Plato  objected  to  calling  a 
point  a  "geometrical  fiction."  He  defined  a  point  as  the  "beginning 
of  a  line"  or  as  "an  indivisible  line,"  and  a  line  as  "length  without 
breadth."  He  called  the  point,  line,  surface,  the  "boundaries"  of 
the  line,  surface,  solid,  respectively.  Many  of  the  definitions  in  Euclid 
are  to  be  ascribed  to  the  Platonic  school.  The  same  is  probably  true 
of  Euclid's  axioms.  Aristotle  refers  to  Plato  the  axiom  that  "equals 
subtracted  from  equals  leave  equals." 

0  One  of  the  greatest  achievements  of  Plato  and  his  school  is  the  in- 
vention of  analysis  as  a  method  of  proof.  To  be  sure,  this  method 
had  been  used  unconsciously  by  Hippocrates  and  others;  but  Plato, 
like  a  true  philosopher,  turned  the  instinctive  logic  into  a  conscious, 
legitimate  method. 

*  "If  any  one  scientific  invention  can  claim  pre-eminence  over  all  others,  I  should 
be  inclined  myself  to  erect  a  monument  to  the  unknown  inventor  of  the  mathe- 
matical point,  as  the  supreme  type  of  that  process  of  abstraction  which  has  been 
a  necessary  condition  of  scientific  work  from  the  very  beginning."  Horace  Lamb's 
Address,  Section  A,  Brit.  Ass'n,  1904. 


GREEK  GEOMETRY  27 

The  terms  synthesis  and  analysis  are  used  in  mathematics  in  a  more 
special  sense  than  in  lop;ic.  In  ancient  mathematics  they  had  a  dif- 
ferent nieaning  from  what  they  now  have.  The  oldest  definition  of 
mathematical  analysis  as  oi)posed  to  synthesis  is  that  given  in  Euclid, 
XIII,  5,  which  in  all  probability  was  framed  by  Eudoxus:  '•  Analysis  is 
the  obtaining  of  the  thing  sought  by  assuming  it  and  so  reasoning  up 
to  an  admitted  truths  synthesis  is  the  obtaining  of  the  thing  sought 
by  reasoning  up  to  the  inference  and  proof  of  it>'  The  analytic  method 
is  not  conclusive,  unless  all  operations  involved  in  it  are  knowTi  to 
be  reversible.  To  remove  all  doubt,  the  Greeks,  as  a  rule,  added  to 
the  analytic  process  a  synthetic  one,  consisting  of  a  revet  sion  of  all 
operations  occurring  in  the  analysis.  Thus  the  aim  of  analysis  was 
to  aid  in  the  discovery  of  synthetic  proofs  or  solutions. 

Plato  is  said  to  have  solved  the  problem  of  the  duplication  of  the 
cube.  But  the  solution  is  open  to  the  very  same  objection  which  he 
made  to  the  solutions  by  Archytas,  Eudoxus,  and  ]VIena:?chmus.  He 
called  their  solutions  not  geometrical,  but  mechanical,  for  they  re- 
quired the  use  of  other  instruments  than  the  ruler  and  compasses. 
He  said  that  thereby  "  the  good  of  geometry  is  set  aside  and  destroyed, 
for  M-e  again  reduce  it  to  the  world  of  sense,  instead  of  ele\'ating  and 
imbuing  it  with  the  eternal  and  incorporeal  images  of  thought,  even 
as  it  is  employed  by  God,  for  which  reason  He  always  is  God."  These 
objections  indicate  either  that  the  solution  is  wrongly  attributed  to 
Plato  or  that  he  wished  to  show  how  easily  non-geometric  solutions 
of  that  character  can  be  found.  It  is  now  rigorously  established  that 
the  duplication  problem,  as  well  as  the  trisection  and  quadrature 
problems,  cannot  be  solved  by  means  of  the  ruler  and  compasses 
only. 

Plato  gave  a  healthful  stimulus  to  the  study  of  stereometry,  which 
until  his  time  had  been  entirely  neglected  by  the  Greeks.  The  sphere 
and  the  regular  solids  have  been  studied  to  some  extent,  but  the  prism, 
{pyramid,  cylinder,  and  cone  were  hardly  known  to  exist.  All  these 
solids  became  the  subjects  of  investigation  by  the  Platonic  school. 
One  result  of  these  inquiries  was  epoch-making.  Menaechmus,  an 
associate  of  Plato  and  pupil  of  Eudoxus,  invented  the  conic  sections, 
which,  in  course  of  only  a  century,  raised  geometry  to  the  loftiest  height 
which  it  was  destined  to  reach  during  antiquity.  Menaechmus  cut 
three  kinds  of  cones,  the  "right-angled,"  "acute-angled,"  and  "obtuse- 
angled,"  by  planes  at  right  angles  to  a  side  of  the  cones,  and  thus 
obtained  the  three  sections  which  we  now  call  the  parabola,  ellipse, 
and  h\']:)erbola.  Judging  from  the  two  very  elegant  solutions  of  the 
"Delian  Problem"  by  means  of  intersections  of  these  curves,  Menaech- 
mus must  have  succeeded  well  in  investigating  their  ])roi)erties.  In 
what  manner  he  carried  out  the  graphic  construction  of  these  curves 
is  not  known. 

Another  great  geometer  was  Dinostratus,  the  brother  of  Menaech- 


2S  A  HISTORY  OF  MATHEMATICS 

mus  and  pupil  of  Plato.  Celebrated  is  his  mechanical  solution  of  the 
quadrature  of  the  circle,  by  means  of  the  quadratrix  of  Hippias. 

Perhaps  the  most  brilliant  mathematician  of  this  period  was 
Eudoxus.  He  was  born  at  Cnidus  about  40S  b.  c,  studied  under 
Archytas,  and  later,  for  two  months,  under  Plato.  He  was  imbued 
with  a  true  spirit  of  scientific  inquiry,  and  has  been  called  the  father 
of  scientific  astronomical  observation.  From  the  fragmentary  notices 
of  his  astronomical  researches,  found  in  later  writers,  Ideler  and 
SchiapareUi  succeeded  in  reconstructing  the  system  of  Eudoxus  with 
its  celebrated  representation  of  planetary  motions  by  "concentric 
spheres."  Eudoxus  had  a  school  at  Cyzicus,  went  with  his  pupils  to 
Athens,  visiting  Plato,  and  then  returned  to  Cyzicus,  where  he  died 
355  B.  c.  The  fame  of  the  academy  of  Plato  is  to  a  large  extent  due 
to  Eudoxus's  pupils  of  the  school  at  Cyzicus,  among  whom  are  Men- 
aechmus,  Dinostratus,  Athenaeus,  and  Helicon.  Diogenes  Laertius  de- 
scribes Eudoxus  as  astronomer,  physician,  legislator,  as  well  as  geom- 
eter. The  Eiidemian  Summary  says  that  Eudoxus  "first  increased  the 
number  of  general  theorems,  added  to  the  three  proportions  three 
more,  and  raised  to  a  considerable  quantity  the  learning,  begun  by 
Plato,  on  the  subject  of  the  section,  to  which  he  applied  the  analytical 
method."  By  this  "  section "  is  meant,  no  doubt,  the  "golden  section " 
{sectio  aurea),  which  cuts  a  line  in  extreme  and  mean  ratio.  The  first 
five  propositions  in  Euclid  XIII  relate  to  lines  cut  by  this  section,  and 
are  generally  attributed  to  Eudoxus.  Eudoxus  added  much  to  the 
knowledge  of  solid  geometry.  He  proved,  says  Archimedes,  that  a 
pyramid  is  exactly  one-third  of  a  prism,  and  a  cone  one-third  of  a 
cylinder,  having  equal  base  and  altitude.  The  proof  that  spheres  are 
to  each  other  as  the  cubes  of  their  radii  is  probably  due  to  him.  He 
made  frequent  and  skilful  use  of  the  method  of  exhaustion,  of  which 
he  was  in  all  probability  the  inventor.  A  scholiast  on  Euclid,  thought 
to  be  Proclus,  says  further  that  Eudoxus  practically  invented  the 
whole  of  Euclid's  fifth  book.  Eudoxus  also  found  two  mean  propor- 
tionals between  two  given  lines,  but  the  method  of  solution  is  not 
known. 

Plato  has  been  called  a  maker  of  mathematicians.  Besides  the 
pupils  already  named,  the  Etidemian  Summary  mentions  the  following* 
Theaetetus  of  Athens,  a  man  of  great  natural  gifts,  to  whom,  no  doubt, 
Euclid  was  greatly  indebted  in  the  composition  of  the  loth  book,i 
treating  of  incommensurables  and  of  the  13th  book;  Leodamas  of 
Thasos;  Neocleides  and  his  pupil  Leon,  who  added  much  to  the  work 
of  their  predecessors,  for  Leon  wrote  an  Elements  carefully  designed, 
both  in  number  and  utility  of  its  proofs;  Theudius  of  Magnesia,  who 
composed  a  very  good  book  of  Elements  and  generalised  propositions, 
which  had  been  confined  to  particular  cases;  Hermotimus  of  Col- 
ophon, who  discovered  many  propositions  of  the  Elements  and  com- 
1  G.  J.  Allman,  op.  cit.,  p.  212. 


GREEK  GEOMETRY  29 

posed  some  on  loci;  and,  finally,  the  names  of  Amyclas  of  Heraclea, 
Cyzicenus  of  Athens,  and  Philippus  of  Mende. 

A  skilful  mathematician  of  whose  life  and  works  we  have  no  details 
is  Aristaeus,  the  elder,  probably  a  senior  contemporary  of  Euclid.  The 
fact  that  he  wrote  a  work  on  conic  sections  tends  to  show  that  much 
progress  had  been  made  in  their  study  during  the  time  of  Menaechmus. 
Aristaeus  wrote  also  on  regular  solids  and  cultivated  the  analytic 
method.  His  works  contained  probably  a  summary  of  the  researches 
of  the  Platonic  school.^ 

Aristotle  (3S4-322  b.  c),  the  systematiser  of  deductive  logic,  though 
not  a  professed  mathematician,  promoted  the  science  of  geometry  by 
improving  some  of  the  most  difficult  definitions.  His  Physics  contains 
passages  with  suggestive  hints  of  the  principle  of  virtual  velocities. 
He  gave  the  best  discussion  of  continuity  and  of  Zeno's  arguments 
against  motion,  found  in  antiquity.  About  his  time  there  appeared  a 
work  called  Mechanica,  of  which  he  is  regarded  by  some  as  the  author. 
JSIechanics  ^vas  totally  neglected  by  the  Platonic  school. 

The  First  Alexandrian  School 

In  the  previous  pages  we  have  seen  the  birth  of  geometry  in  Egypt, 
its  transference  to  the  Ionian  Islands,  thence  to  Lower  Italy  and  to 
Athens.  We  have  witnessed  its  growth  in  Greece  from  feeble  child- 
hood to  \agorous  manhood,  and  now  we  shall  see  it  return  to  the  land 
of  its  birth  and  there  derive  new  vigor. 

During  her  declining  years,  immediately  following  the  Pelopon- 
nesian  War,  Athens  produced  the  greatest  scientists  and  philosophers 
of  antiquity.  It  was  the  time  of  Plato  and  Aristotle.  In  338  b.  c,  at 
the  battle  of  Cha^ronea,  Athens  was  beaten  by  Philip  of  Macedon, 
and  her  power  was  broken  forever.  Soon  after,  Alexander  the  Great, 
the  son  of  Philip,  started  out  to  conquer  the  world.  In  eleven  years 
he  built  up  a  great  empire  which  broke  to  pieces  in  a  day.  Eg>']:)t 
fell  to  the  lot  of  Ptolemy  Soter.  Alexander  had  founded  the  seaport 
of  Alexandria,  which  soon  became  the  "  noblest  of  all  cities."  Ptolemy 
made  Alexandria  the  capital.  The  history  of  Egypt  during  the  next 
three  centuries  is  mainly  the  history  of  Alexandria.  Literature, 
philosophy,  and  art  were  diligently  cultivated.  Ptolemy  created  the 
university  of  Alexandria.  He  founded  the  great  Library  and  built 
laboratories,  museums,  a  zoological  garden,  and  promenades.  Alex- 
andria soon  became  the  great  centre  of  learning. 

Demetrius  Phalereus  was  invited  from  Athens  to  take  charge  of  the 
Library,  and  it  is  probable,  says  Gow,  that  Euclid  was  invited  with 
him  to  open  the  mathematical  school.  According  to  the  studies  of 
H.  Vogt,2  Euclid  was  bom  about  365  b.  c.  and  wrote  his  Elements 

'  G.  J.  Allman,  op.  cit.,  p.  205. 

^  Bibliolheca  mathcmalica,  3  S.,  Vul.  ij,  1913,  {jp.  193-202. 


30  A  HISTORY  OF  MATHEMATICS 

between  330  and  320  b.  c.  Of  the  life  of  Euclid,  little  is  known,  except 
what  is  added  by  Proclus  to  the  Eudemian  Summary.  Euclid,  says 
Proclus,  was  younger  than  Plato  and  older  than  Eratosthenes  and 
Archimedes,  the  latter  of  whom  mentions  him.  He  was  of  the  Platonic 
sect,  and  well  read  in  its  doctrines.  He  collected  the  EJcments,  put 
in  order  much  that  Eudoxus  had  prepared,  completed  many  things  of 
Thesetetus,  and  was  the  first  who  reduced  to  unobjectionable  demon- 
stration the  imperfect  attempts  of  his  predecessors.  When  Ptolemy 
once  asked  him  if  geometry  could  not  be  mastered  by  an  easier  process 
than  by  studying  the  Elements,  Euclid  returned  the  answer,  "There 
is  no  royal  roid  to  geometry."  Pappus  states  that  Euclid  was  distin- 
guished by  the  fairness  and  kindness  of  his  disposition,  particularly 
toward  those  who  could  do  anything  to  advance  the  mathematical 
sciences.  Pappus  is  evidently  making  a  contrast  to  Apollonius,  of 
whom  he  more  than  insinuates  the  opposite  character.^  A  pretty 
little  story  is  related  by  Stobasus:  -  "A  youth  who  had  begun  to  read 
geometry  with  Euclid,  when  he  had  learnt  the  first  proposition,  in- 
quired, 'What  do  I  get  by  learning  these  things.^'  So  Euclid  called 
his  slave  and  said,  'Give  him  threepence,  since  he  must  make  gain 
out  of  what  he  learns.'"  These  are  about  all  the  personal  details 
preserved  by  Greek  writers.  Syrian  and  Arabian  writers  claini  to 
know  much  more,  but  they  are  unreliable.  At  one  time  Euclid  of 
Alexandria  was  universally  confounded  with  Euclid  of  Megara,  who 
lived  a  century  earlier. 

The  fame  of  Euclid  has  at  all  times  rested  mainly  upon  his  book  on 
geometry,  called  the  Elements.  This  book  was  so  far  superior  to  the 
Elements  written  by  Hippocrates,  Leon,  and  Theudius,  that  the  latter 
works  soon  perished  in  the  struggle  for  existence.  The  Greeks  gave 
Euclid  the  special  title  of  "the  author  of  the  Elements.''^  It  is  a  re- 
markable fact  in  the  history  of  geometry,  that  the  Elements  of  Euclid, 
written  over  two  thousand  years  ago,  are  still  regarded  by  some  as  the 
best  introduction  to  the  mathematical  sciences.  In  England  they 
were  used  until  the  present  century  extensively  as  a  text-book  in 
schools.  Some  editors  of  Euclid  have,  however,  been  inclined  to  credit 
him  with  more  than  is  his  due.  They  would  have  us  believe  that  a 
finished  and  unassailable  system  of  geometry  sprang  at  once  from  the 
brain  of  Euclid,  "an  armed  Minerva  from  the  head  of  Jupiter."  They 
fail  to  mention  the  earlier  eminent  mathematicians  from  whom  Euclid 
got  his  material.  Comparatively  few  of  the  propositions  and  proofs 
in  the  Elements  are  his  own  discoveries.  In  fact,  the  proof  of  the 
"Theorem  of  Pythagoras"  is  the  only  one  directly  ascribed  to  him. 
AUman  conjectures  that  the  substance  of  Books  I,  II,  IV  comes  from 
the  Pythagoreans,  that  the  substance  of  Book  VI  is  due  to  the  Pytha- 

'  A.  De  Morgan,  "  Eucleides  "  in  Smith's  Dictionary  of  Greek  and  Roman  Biography 
and  Mythology. 

2  J.  Gow,  op.  cil.,  p.  195. 


GREEK  GEOMETRY 


31 


goreans  and  Eudoxus,  the  latter  contributing  the  doctrine  of  propor- 
tion as  applicable  to  incommensurables  and  also  the  Method  of  Ex- 
haustions (Book  XII),  that  Theastetus  contributed  much  toward 
Books  X  and  XIII,  that  the  principal  part  of  the  original  work  of 
Euclid  himself  is  to  be  found  in  Book  X}  Euclid  was  the  greatest 
systematiser  of  his  time.  By  careful  selection  from  the  material  before 
him,  and  by  logical  arrangement  of  the  propositions  selected,  he  built 
up,  from  a  few  definitions  and  axioms,  a  proud  and  lofty  structure. 
It  would  be  erroneous  to  believe  that  he  incorporated  into  his  Elements 
all  the  elementary  theorems  known  at  his  time.  Archimedes,  Apol- 
lonius,  and  even  he  himself  refer  to  theorems  not  included  in  his  Ele- 
ments, as  being  well-known  truths. 

The  text  of  the  Elements  that  was  commonly  used  in  schools  was 
Theon's  edition.  Theon  of  Alexandria,  the  father  of  Hypatia,  brought 
out  an  edition,  about  700  years  after  Euclid,  with  some  alterations  in 
the  text.  As  a  consequence,  later  commentators,  especially  Robert 
Simson,  who  labored  under  the  idea  that  Euclid  must  be  absolutely 
perfect,  made  Theon  the  scapegoat  for  all  the  defects  which  they 
thought  they  could  discover  in  the  text  as  they  knew  it.  But  among 
the  manuscripts  sent  by  Napoleon  I  from  the  Vatican  to  Paris  was 
found  a  copy  of  the  Elements  believed  to  be  anterior  to  Theon's  recen- 
sion. Many  variations  from  Theon's  version  were  noticed  therein, 
but  they  were  not  at  all  important,  and  showed  that  Theon  generally 
made  only  verbal  changes.  The  defects  in  the  Elements  for  which 
Theon  was  blamed  must,  therefore,  be  due  to  Euclid  himself.  The 
Elements  used  to  be  considered  as  offering  models  of  scrupulously 
rigorous  demonstrations.  It  is  certainly  true  that  in  point  of  rigor 
it  compares  favorably  with  its  modern  rivals;  but  when  examined 
in  the  light  of  strict  mathematical  logic,  it  has  been  pronounced  by 
C.  S.  Peirce  to  be  "riddled  with  fallacies."  The  results  are  correct 
only  because  the  writer's  experience  keeps  him  on  his  guard.  In 
many  proofs  Euclid  relies  partly  upon  intuition. 

At  the  beginning  of  our  editions  of  the  Elements,  under  the  head  of 
definitions,  are  given  the  assumptions  of  such  notions  as  the  point, 
line,  etc.,  and  some  verbal  explanations.  Then  follow  three  postulates 
or  demands,  and  twelve  axioms.  The  term  "axiom"  was  used  by 
Proclus,  but  not  by  Euclid.  He  speaks,  instead,  of  "common  no- 
tions"— common  either  to  all  men  or  to  all  sciences.  There  has  been 
much  controversy  among  ancient  and  modern  critics  on  the  postulates 
and  axioms.  An  immense  preponderance  of  manuscripts  and  the 
testimony  of  Proclus  place  the  "axioms"  about  right  angles  and 
parallels  among  the  postulates.-    This  is  indeed  their  proper  place, 

^  G.  J.  Allman,  op.  cil.,  p.  211. 

^  A.  De  Morgan,  loc.  cil.;  H.  Hankcl,  Thcorie  dcr  Complcxcn  Zahlcnsystane,  Leip- 
zig, 1867,  p.  52.  In  the  various  editions  of  Euclid's  Elemenls  dilTerent  numbers  are 
assigned  to  the  axioms.    Thus  the  parallel  axiom  is  called  by  Robert  Simson  the 


32  A  HISTORY  OF  MATHEMATICS 

for  they  u.yc  really  assumptions,  and  not  common  notions  or  axioms. 
The  postulate  about  parallels  plays  an  important  role  in  the  history 
of  non-Euclidean  geometry.  An  important  postulate  which  Euclid 
missed  was  the  one  of  superposition,  according  to  which  figures  can 
be  moved  about  in  space  without  any  alteration  in  form  or  magnitude. 
The  Elements  contains  thirteen  books  by  Euclid,  and  two,  of  which 
it  is  supposed  that  Hypsicles  and  Damascius  are  the  authors.  The 
first  four  books  are  on  plane  geometry.  The  fifth  book  treats  of  the 
theory  of  proportion  as  applied  to  magnitudes  in  general.  It  has  been 
greatly  admired  because  of  its  rigor  of  treatment.  Beginners  find  the 
book  difficult.  Expressed  in  modern  symbols,  Euclid's  definition  of 
proportion  is  thus:  Four  magnitudes,  a,  h,  c,  d,  are  in  proportion,  when 

for  any  integers  ni  and  n,  we  have  simultaneously  ma=nb,  and  mc 

nd.  Says  T.  L.  Heath,i  "certain  it  is  that  there  is  an  exact  corre- 
spondence, almost  coincidence,  between  Euclid's  definition  of  equal 
ratios  and  the  modern  theory  of  irrationals  due  to  Dedekind.  H.  G. 
Zeuthen  finds  a  close  resemblance  between  Euclid's  definition  and 
Weierstrass'  definition  of  equal  numbers.  The  sixth  book  develops 
the  geometry  of  similar  figures.  Its  27th  Proposition  is  the  earliest 
maximum  theorem  known  to  history.  The  seventh,  eighth,  ninth 
books  are  on  the  theory  of  numbers,  or  on  arithmetic.  According  to 
P.  Tannery,  the  knowledge  of  the  existence  of  irrationals  must  have 
greatly  affected  the  mode  of  writing  the  Elements.  The  old  naive 
theory  of  proportion  being  recognized  as  untenable,  proportions 
are  not  used  at  all  in  the  first  four  books.  The  rigorous  theory  of 
Eudoxus  was  postponed  as  long  as  possible,  because  of  its  difficulty. 
The  interpolation  of  the  arithmetical  books  VII-IX  is  explained 
as  a  preparation  for  the  fuller  treatment  of  the  irrational  in  book  X. 
Book  VII  explains  the  G.  C.  D.  of  two  numbers  by  the  process 
of  division  (the  so-called  "Euclidean  method").  The  theory  of 
proportion  of  (rational)  numbers  is  then  developed  on  the  basis  of 
the  definition,  "Numbers  are  proportional  when  the  first  is  the  same 
multiple,  part,  or  parts  of  the  second  that  the  third  is  of  the  fourth." 
This  is  believed  to  be  the  older,  Pythagorean  theory  of  proportion. 2 
The  tenth  treats  of  the  theory  of  incommensurables.  De  Morgan  con- 
sidered this  the  most  wonderful  of  all.  We  give  a  fuller  account  of  it 
under  the  head  of  Greek  Arithmetic.    The  next  three  books  are  on 

1 2th,  by  Bolyai  the  nth,  by  Clavius  the  13th,  by  F.  Peyrard  the  sth.  It  is  called 
the  5th  postulate  in  old  manuscripts,  also  by  Heiberg  and  Menge  in  their  annotated 
edition  of  Euclid's  works,  in  Greek  and  Latin,  Leipzig,  1883,  and  by  T.  L.  Heath 
in  his  Thirteen  Books  of  Euclid's  Elements,  Vols.  I-III,  Cambridge,  1908.  Heath's 
is  the  most  recent  translation  into  English  and  is  very  fully  and  ably  annotated. 

1  T.  L.  Heath,  op.  cil..  Vol.  II,  p.  124. 

2  Read  H.  B.  Fine,  "Ratio,  Proportion  and  Measurement  in  the  Elements  of 
Euclid,"  Annals  of  Mathematics,  Vol.  XIX,  1917,  pp.  70-76. 


GREEK  GEOMETRY 


33 


stereometry.  The  eleventh  contains  its  more  elementary  theorems; 
the  twelfth,  the  metrical  relations  of  the  pyramid,  prism,  cone,  cylinder, 
and  sphere.  The  thirteenth  treats  of  the  regular  polygons,  especially 
of  the  triangle  and  pentagon,  and  then  uses  them  as  faces  of  the  five 
tegular  solids;  namely,  the  tetraedron,  octaedron,  icosaedron,  cube, 
and  dodecaedron.  The  regular  solids  were  studied  so  extensively  by 
the  Platonists  that  they  received  the  name  of  "  Platonic  figures."  The 
statement  of  Proclus  that  the  whole  aim  of  Euclid  in  writing  the  Ele- 
ments  was  to  arrive  at  the  construction  of  the  regular  solids,  is  ob- 
viously wTong.  The  fourteenth  and  fifteenth  books,  treating  of  solid 
geometry,  are  apocryphal.  It  is  interesting  to  see  that  to  Euclid,  and 
to  Greek  mathematicians  in  general,  the  existence  of  areas  was  evident 
from  intuition.  The  notion  of  non-quadrable  areas  had  not  occurred 
to  them. 

A  remarkable  feature  of  Euclid's,  and  of  all  Greek  geometry  before 
Archimedes  is  that  it  eschews  mensuration.  Thus  the  theorem  that 
the  area  of  a  triangle  equals  half  the  product  of  its  base  and  its  altitude 
is  foreign  to  Euclid. 

Another  extant  book  of  Euclid  is  the  Data.  It  seems  to  have  been 
written  for  those  who,  having  completed  the  Elements,  wish  to  acquire 
the  power  of  solving  new  problems  proposed  to  them.  The  Data  is 
a  course  of  practice  in  analysis.  It  contains  little  or  nothing  that  an 
intelligent  student  could  not  pick  up  from  the  Elements  itself.  Hence 
it  contributes  little  to  the  stock  of  scientific  knowledge.  The  following 
are  the  other  works  with  texts  more  or  less  complete  and  generally 
attributed  to  Euclid:  Phoenomena,  a  work  on  spherical  geometry  and 
astronomy;  Optics,  which  develops  the  hypothesis  that  light  proceeds 
from  the  eye,  and  not  from  the  object  seen;  Catoptrica,  containing 
propositions  on  reflections  from  mirrors:  De  Divisionibtis,  a  treatise  on 
the  division  of  plane  figures  into  parts  having  to  one  another  a  given 
ratio;  i  Scctio  Canonis,  a  work  on  musical  intervals.  His  treatise  on 
Porisms  is  lost;  but  much  learning  has  been  expended  by  Robert  Sim- 
son  and  M.  Chasles  in  restoring  it  from  numerous  notes  found  in  the 
\\Titings  of  Pappus.  The  term  "porism"  is  vague  in  meaning.  Ac- 
cording to  Proclus,  the  aim  of  a  porism  is  not  to  state  some  property 
or  truth,  like  a  theorem,  nor  to  effect  a  construction,  like  a  problem, 
but  to  find  and  bring  to  view  a  thing  which  necessarily  exists  with 
given  numbers  or  a  given  construction,  as,  to  find  the  centre  of  a  given 
circle,  or  to  find  the  G.  C.  D.  of  two  given  numbers.  Porisms,  ac- 
cording to  Chasles,  are  incomplete  theorems,  "expressing  certain 
relations  between  things  variable  according  to  a  common  law." 
Euclid's  other  lost  works  are  Fallacies,  containing  exercises  in  detec- 
tion of  fallacies;  Conic  Sections,  in  four  books,  which  are  the  foundation 
of  a  work  on  the  same  subject  by  Apollonius;  and  Loci  on  a  Surface, 

'  A  careful  restoration  was  brought  out  in  1915  by  R.  C.  Archibald  of  Brown 
University. 


34  A  HISTORY  OF  MATHEMATICS 

the  meaning  of  which  title  is  not  understood.  Heiberg  believes  it  to 
mean  "loci  which  are  surfaces." 

The  immediate  successors  of  Euclid  in  the  mathematical  school  at 
Alexandria  were  probably  Conon,  Dositheus,  and  Zeuxippus,  but 
little  is  knoA\Ti  of  them. 

Archimedes  (287?-2i2  b.  c),  the  greatest  mathematician  of  an- 
tiquity, was  born  in  Syracuse.  Plutarch  calls  him  a  relation  of  King 
Hieron;  but  more  reliable  is  the  statement  of  Cicero,  who  tdls  us  he 
was  of  low  birth.  Diodorus  says  he  visited  Eg^pt,  and,  since  he  was 
a  great  friend  of  Conon  and  Eratosthenes,  it  is  highly  probable  that 
he  studied  in  Alexandria.  This  belief  is  strengthened  by  the  fact  that 
he  had  the  most  thorough  acquaintance  with  all  the  work  previously 
done  in  mathematics.  He  returned,  however,  to  S>Tacuse,  where  he 
made  himself  useful  to  his  admiring  friend  and  patron.  King  Hieron, 
by  applying  his  extraordinary  inventive  genius  to  the  construction  of 
various  war-engines,  by  which  he  inflicted  much  loss  on  the  Romans 
during  the  siege  of  Marcellus.  The  story  that,  by  the  use  of  mirrors 
reflecting  the  sun's  rays,  he  set  on  fire  the  Roman  ships,  w^hen  they 
came  within  bow-shot  of  the  walls,  is  probably  a  fiction.  The  city 
was  taken  at  length  by  the  Romans,  and  Archimedes  perished  in  the 
indiscriminate  slaughter  which  followed.  According  to  tradition,  'he 
X^s,  at  the  time,  studying  the  diagram  to  some  problem  drawm  in  the 
sand.  As  a  Roman  soldier  approached  him,  he  called  out,  "Don't  spoil 
my  circles."  The  soldier,  feeling  insulted,  rushed  upon  him  and  killed 
him.  No  blame  attaches  to  the  Roman  general  ilarcellus,  who  ad- 
mired his  genius,  and  raised  in  his  honor  a  tomb  bearing  the  figure 
of  a  sphere  inscribed  in  a  C3-linder.  When  Cicero  was  in  Syracuse, 
he  found  the  tomb  buried  under  rubbish. 

Archimedes  was  admired  by  his  fellow-citizens  chiefly  for  his  me- 
chanical inventions;  he  himself  prized  far  more  highly  his  discoveries 
in  pure  science.  He  declared  that  "every  kind  of  art  which  was  con- 
nected with  daily  needs  was  ignoble  and  vulgar."  Some  of  his  works 
have  been  lost.  The  following  are  the  extant  books,  arranged  ap- 
proximately in  chronological  order:  i.  Two  books  on  Eqiiiporiderance 
of  Planes  or  Centres  of  Plane  Cravilies,  between  w^hich  is  inserted  his 
treatise  on  the  Quadrature  of  the  Parabola;  2.  The  Method;  3.  Two  books 
on  the  Sphere  and  Cylinder;  4.  The  Measurement  of  the  Circle;  5.  On 
Spirals;  6.  Conoids  and  Spheroids;  7.  The  Sand-Counter;  8.  Two  books 
on  Floating  Bodies;  9.  Fifteen  Lemmas. 

In  the  book  on  the  Measurement  of  the  Circle,  Archimedes  proves 
first  that  the  area  of  a  circle  is  equal  to  that  of  a  right  triangle  having 
the  length  of  the  circumference  for  its  base,  and  the  radius  for  its 
altitude.  In  this  he  assumes  that  there  exists  a  straight  line  equal  in 
length  to  the  circumference — an  assumption  objected  to  by  some 
ancient  critics,  on  the  ground  that  it  is  not  evident  that  a  straight 
line  can  equal  a  curved  one.    The  finding  of  such  a  line  was  the  next 


GREEK  GEOMEFRY  35 

problem.  He  first  finds  an  upper  limit  to  the  ratio  of  the  circumfer- 
ence to  the  diameter,  or  ir.  To  do  this,  he  starts  with  an  equilateral 
triangle  of  which  the  base  is  a  tangent  and  the  vertex  is  the  centre  of 
the  circle.  By  successively  bisecting  the  angle  at  the  centre,  by  com- 
paring ratios,  and  by  taking  the  irrational  scjuare  roots  always  a  little 
too  small,  he  finally  arrived  at  the  conclusion  that  7r<3y.  Next  he 
finds  a  lower  limit  by  inscribing  in  the  circle  regular  polygons  of  6,  12, 
24,  48,  96  sides,  finding  for  each  successive  polygon  its  perimeter, 
which  is,  of  course,  always  less  than  the  circumference.  Thus  he 
finally  concludes  that  ''the  circumference  of  a  circle  exceeds  three 
times  its  diameter  by  a  part  which  is  less  than  i  but  more  than  ^ 
of  the  diameter."  This  approximation  is  exact  enough  for  most  pur- 
poses. 

The  Quadrature  of  the  Parabola  contains  two  solutions  to  the  prob- 
lem— one  mechanical,  the  other  geometrical.  The  method  of  ex- 
haustion is  used  in  both. 

It  is  noteworthy  that,  perhaps  through  the  influence  of  Zeno,  in- 
finitesimals (infinitely  small  constants)  were  not  used  in  rigorous 
demonstration.  In  fact,  the  great  geometers  of  the  period  now  under 
consideration  resorted  to  the  radical  measure  of  excluding  them  from 
demonstrative  geometry  by  a  postulate.  This  was  done  by  Eudoxus, 
Euclid,  and  Archimedes.  In  the  preface  to  the  Guadrature  of  the  Parab- 
ola, occurs  the  so-called  "Archimedean  postulate,"  which  Archimedes 
himself  attributes  to  Eudoxus:  "When  two  spaces  are  unequal,  it  is 
possible  to  add  to  itself  the  difference  by  which  the  lesser  is  surpassed 
by  the  greater,  so  often  that  every  finite  space  will  be  exceeded." 
Euclid  (Elements  V,  4)  gives  the  postulate  in  the  form  of  a  definition: 
"Magnitudes  are  said  to  have  a  ratio  to  one  another,  when  the  less 
can  be  multiplied  so  as  to  exceed  the  other."  Nevertheless,  infinitesi- 
mals may  have  been  used  in  tentative  researches.  That  such  was  the 
case  with  Archimedes  is  evident  from  his  book.  The  Method,  formerly 
thought  to  be  irretrievably  lost,  but  fortunately  discovered  by  Heiberg 
in  1906  in  Constantinople.  The  contents  of  this  book  shows  that  he 
considered  infinitesimals  sufiiciently  scientific  to  suggest  the  truths  of 
theorems,  but  not  to  furnish  rigorous  proofs.  In  finding  the  areas  of 
])arabolic  segments,  the  volumes  of  spherical  segments  and  other  solids 
of  revolution,  he  uses  a  mechanical  process,  consisting  of  the  weighing 
of  infinitesimal  elements,  which  he  calls  straight  lines  or  })lane  areas, 
but  which  are  really  infinitely  narrow  stri[)S  or  infinitely  thin  plane 
laminae.^  The  breadth  or  thickness  is  regarded  as  being  the  same  in 
the  elements  weighed  at  any  one  time.  The  Archimedean  postulate 
did  not  command  the  interest  of  mathematicians  until  the  modern 
arithmetic  continuum  was  created.  It  was  O.  Stolz  that  showed  that 
it  was  a  ^;onscquencL'  of  Dedekind's  postulate  relating  to  "sections." 

^  T.  L.  Heath,  Mdhod  of  Archimedes,  Cambridge,  1912,  p.  8. 


36  A  HISTORY  OF  MATHEMATICS 

It  would  seem  that,  in  his  great  researches,  Archimedes'  mode  of 
procedure  was,  to  start  with  mechanics  (centre  of  mass  of  surfaces  and 
solids)  and  by  his  infinitesimal-mechanical  method  to  discover  new 
results  for  which  later  he  deduced  and  published  the  rigorous  proofs. 
Archimedes  knew  the  integral  ^  JxHx. 

Archimedes  studied  also  the  ellipse  and  accomplished  its  quadrature, 
but  to  the  hyperbola  he  seems  to  have  paid  less  attention.  It  is  be- 
lieved that  he  wrote  a  book  on  conic  sections. 

Of  all  his  discoveries  Archimedes  prized  most  highly  those  in  his 
Sphere  aiid  Cylinder.  In  it  are  proved  the  new  theorems,  that  the 
surface  of  a  sphere  is  equal  to  four  times  a  great  circle;  that  the  surface 
segment  of  a  sphere  is  equal  to  a  circle  whose  radius  is  the  straight 
line  drawn  from  the  vertex  of  the  segment  to  the  circumference  of  its 
basal  circle;  that  the  volume  and  the  surface  of  a  sphere  are  |  of  the 
volume  and  surface,  respectively,  of  the  cylinder  circumscribed  about 
the  sphere.  Archimedes  desired  that  the  figure  to  the  last  proposition 
be  inscribed  on  his  tomb.    This  was  ordered  done  by  Alarcellus. 

The  spiral  now  called  the  "spiral  of  Archimedes,"  and  described  in 
the  book  On  Spirals,  was  discovered  by  Archimedes,  and  not,  as  some 
believe,  by  his  friend  Conon.-  His  treatise  thereon  is,  perhaps,  the 
most  wonderful  of  all  his  works.  Nowadays,  subjects  of  this  kind 
are  made  easy  by  the  use  of  the  infinitesimal  calculus.  In  its  stead 
the  ancients  used  the  method  of  exhaustion.  Nowhere  is  the  fertility 
of  his  genius  more  grandly  displayed  than  in  his  masterly  use  of  this 
method.  With  Euclid  and  his  predecessors  the  method  of  exhaustion 
was  only  the  means  of  proving  propositions  which  must  have  been 
seen  and  believed  before  they  were  proved.  But  in  the  hands  of 
Archimedes  this  method,  perhaps  combined  with  his  infinitesunal- 
mechanical  method,  became  an  instrument  of  discovery. 

By  the  word  "conoid,"  in  his  book  on  Conoids  and  Spheroids,  is 
meant  the  solid  produced  by  the  revolution  of  a  parabola  or  a  hyper- 
bola about  its  axis.  Spheroids  are  produced  by  the  revolution  of  an 
ellipse,  and  are  long  or  flat,  according  as  the  ellipse  revolves  around 
the  major  or  minor  axis.    The  book  leads  up  to  the  cubature  of  these 

solids.  A  few  constructions  of  geo- 
metric figures  were  given  by  Archi- 
medes and  Appolonius  which  were 
effected  by  "insertions."  In  the 
following  trisection  of  an  angle,  at- 
tributed by  the  Arabs  to  Archi- 
medes, the  "insertion"  is  achieved  by  the  aid  of  a  graduated  ruler.3 
To  trisect  the  angle  CAB,  draw  the  arc  BCD.    Then  "insert"  the 

1  H.  G.  Zeuthen  in  Bibliolheca  mathematica,  3  S.,  Vol.  7,  1906-7,  p.  347. 
-  M.  Cantor,  op.  cit.,  Vol.  I,  3  Aufl.,  1907,  p.  306. 

'  F.  Enriques,  Fragen  der  Elemenlargeometrie,  deutsche  Ausg.  v.  H.  Fleischer,  II, 
Leipzig,  1907,  p.  234. 


GREEK  GEOMETRY  37 

distance  FE,  equal  to  AB,  marked  on  an  edge  passing  through  C 
and  moved  until  the  points  E  and  F  are  located  as  shown  in  the 
figure.    The  required  angle  is  EFD. 

His  arithmetical  treatise  and  problems  will  be  considered  later. 
We  shall  now  notice  his  works  on  mechanics.  Archimedes  is  the 
author  of  the  first  sound  knowledge  on  this  subject.  Archytas,  Aris- 
totle, and  others  attempted  to  form  the  known  mechanical  truths  into 
a  science,  but  failed.  Aristotle  knew  the  property  of  the  lever,  but 
could  not  establish  its  true  mathematical  theory.  The  radical  and 
fatal  defect  in  the  speculations  of  the  Greeks,  in  the  opinion  of  Whewell, 
was  "that  though  they  had  in  their  possession  facts  and  ideas,  the 
ideas  were  not  distinct  and  appropriate  to  the  facts."  For  instance, 
Aristotle  asserted  that  when  a  body  at  the  end  of  a  lever  is  moving, 
it  may  be  considered  as  having  two  motions;  one  in  the  direction  of 
the  tangent  and  one  in  the  direction  of  the  radius;  the  former  motion 
is,  he  says,  according  to  nature,  the  latter  contrary  to  nature.  These 
inappropriate  notions  of  "natural"  and  "unnatural"  motions,  to- 
gether with  the  habits  of  thought  which  dictated  these  speculations, 
made  the  perception  of  the  true  grounds  of  mechanical  properties 
impossible.^  It  seems  strange  that  even  after  Archimedes  had  en- 
tered upon  the  right  path,  this  science  should  have  remained  ab- 
solutely stationary  till  the  time  of  Galileo — a  period  of  nearly  two 
thousand  years. 

The  proof  of  the  property  of  the  lever,  given  in  his  Equi ponder ance 
of  Planes,  holds  its  place  in  many  text-books  to  this  day.  Mach  - 
criticizes  it.  "From  the  mere  assumption  of  the  equilibrium  of  equal 
weights  at  equal  distances  is  derived  the  inverse  proportionality  of 
weight  and  lever  arm!  How  is  that  possible? "  Archimedes'  estimate 
of  the  efficiency  of  the  lever  is  expressed  in  the  saying  attributed  to 
him,  "  Give  me  a  fulcrum  on  which  to  rest,  and  I  will  move  the  earth." 

While  the  Equiponderance  treats  of  solids,  or  ihe  equilibrium  of 
solids,  the  book  on  Floating  Bodies  treats  of  hydrostatics.  His  atten- 
tion was  first  drawn  to  the  subject  of  specific  gravity  when  King  Hieron 
asked  him  to  test  whether  a  crown,  professed  by  the  maker  to  be  pure 
gold,  was  not  alloyed  with  silver.  The  story  goes  that  our  philosopher 
was  in  a  bath  when  the  true  method  of  solution  flashed  on  his  mind. 
He  immediately  ran  home,  naked,  shouting,  "I  have  found  it!"  To 
solve  the  problem,  he  took  a  piece  of  gold  and  a  piece  of  silver,  each 
weighing  the  same  as  the  crown.  According  to  one  author,  he  deter- 
mined the  volume  of  water  displaced  by  the  gold,  silver,  and  croA\'n 
respectively,  and  calculated  from  that  the  amount  of  gold  and  silver 

1  William  Whewell,  History  of  Ihe  Inductive  Sciences,  3rd  Ed.,  New  York,  1858, 
Vol.  I,  p.  87.  William  Whewell  (i 794-1 866)  was  Master  of  Trinity  College,  Cam- 
bridge. 

^  E.  Mach,  The  Science  of  Mechanics,  tr.  by  T.  McCormack,  Chicago,  1907,  p.  14. 
Ernst  Mach  (1838-1916)  was  professor  of  the  history  and  theory  of  the  inductive 
>cience£  at  the  university  of  V^ienna. 


38  A  HISTORY  OF  MATHEMATICS 

in  the  crown.  According  to  another  writer,  he  weighed  separately 
the  gold,  silver,  and  crown,  while  immersed  in  water,  thereby  deter- 
mining their  loss  of  weight  in  water.  From  tJicse  data  he  easily  found 
the  solution.  It  is  possible  that  Archimedes  solved  the  problem  by 
both  methods. 

After  examining  the  writings  of  Archimedes,  one  can  well  under- 
stand how,  in  ancient  times,  an  "Archimedean  problem"  came  to 
mean  a  problem  too  deep  for  ordinary  minds  to  solve,  and  how  an 
"Archimedean  proof"  came  to  be  the  synonym  for  unquestionable 
certainty.  Archimedes  wrote  on  a  very  wide  range  of  subjects,  and 
displayed  great  profundity  in  each.    He  is  the  Newton  of  antiquity. 

Eratosthenes,  eleven  years  younger  than  Archimedes,  was  a  native 
of  Cyrene.  He  was  educated  in  Alexandria  under  Calliniachus  the 
poet,  whom  he  succeeded  as  custodian  of  the  Alexandrian  Library. 
His  many-sided  activity  may  be  inferred  from  his  works.  He  wrote 
on  Good  and  Evil,  Measurement  of  the  Earth,  Comedy,  Geography, 
Chronology,  Constellations,  and  the  Duplication  of  the  Cube.  He  was 
also  a  philologian  and  a  poet.  He  measured  the  obliquity  of  the 
ecliptic  and  invented  a  device  for  finding  prime  numbers,  to  be  de- 
scribed later.  Of  his  geometrical  WTitings  we  possess  only  a  letter  to 
Ptolemy  Euergetes,  giving  a  history  of  the  duplication  problem  and 
also  the  description  of  a  very  ingenious  mechanical  contrivance  of  his 
own  to  solve  it.  In  his  old  age  he  lost  his  eyesight,  and  on  that  account 
is  said  to  have  committed  suicide  by  voluntary  starvation. 

About  forty  years  after  Archimedes  flourished  Apollonius  of  Perga, 
whose  genius  nearly  equalled  that  of  his  great  predecessor.  He  incon- 
testably  occupies  the  second  place  in  distinction  among  ancient  mathe- 
maticians. Apollonius  was  born  in  the  reign  of  Ptolemy  Euergetes 
and  died  under  Ptolemy  Philopator,  who  reigned  222-205  B.  c.  He 
studied  at  Alexandria  under  the  successors  of  Euclid,  and  for  some 
time,  also,  at  Pergamum,  where  he  made  the  acquaintance  of  that 
Eudemus  to  whom  he  dedicated  the  first  three  books  of  his  Conic 
Sections.  The  brilliancy  of  his  great  work  brought  him  the  title  of  the 
"Great  Geometer."    This  is  all  that  is  known  of  his  life. 

His  Conic  Sections  were  in  eight  books,  of  which  the  first  four  only 
have  come  down  to  us  in  the  original  Greek.  The  next  three  books 
were  unknown  in  Europe  till  the  middle  of  the  seventeenth  century, 
when  an  Arabic  translation,  made  about  1250,  was  discovered.  The 
eighth  book  has  never  been  found.  In  17 10  E.  Halley  of  Oxford  pub- 
lished the  Greek  text  of  the  first  four  books  and  a  Latin  translation 
of  the  remaining  three,  together  with  his  conjectural  restoration  of 
the  eighth  book,  founded  on  the  introductory  lemmas  of  Pappus.  The 
first  four  books  contain  little  more  than  the  substance  of  what  earher 
geometers  had  done.  Eutocius  tells  us  that  Heraclides,  in  his  life  of 
Archimedes,  accused  Appolonius  of  having  appropriated,  in  his  Conic 
Sections,  the  unpublished  discoveries  of  that  great  mathematician. 


GREEK  GEOMETRY  39 

It  is  difficult  to  believe  that  this  charge  rests  upon  good  foundation. 
Eutocius  quotes  Geminus  as  replying  that  neither  Archimedes  nor 
ApoUonius  claimed  to  have  invented  the  conic  sections,  but  that 
Apollonius  had  introduced  a  real  improvement.  While  the  first  three 
or  four  books  were  founded  on  the  works  of  Menaechmus,  Aristaeus, 
Euclid,  and  Archimedes,  the  remaining  ones  consisted  almost  entirely 
of  new  matter.  The  first  three  books  were  sent  to  Eudemus  at  inter- 
vals, the  other  books  (after  Eudemus's  death)  to  one  Attalus.  The 
preface  of  the  second  book  is  interesting  as  showing  the  mode  in 
which  Greek  books  were  "published"  at  this  time.  It  reads  thus: 
"I  have  sent  my  son  Apollonius  to  bring  you  (Eudemus)  the  second 
book  of  my  Conies.  Read  it  carefully  and  communicate  it  to  such 
others  as  are  worthy  of  it.  If  Philonides,  the  geometer,  whom  I  intro- 
duced to  you  at  Ephesus,  comes  into  the  neighbourhood  of  Pergamum, 
give  it  to  him  also."  ^ 

The  first  book,  says  Apollonius  in  his  preface  to  it,  "contains  the 
mode  of  producing  the  three  sections  and  the  conjugate  h}'perbolas 
and  their  principal  characteristics,  more  fully  and  generally  worked 
out  than  in  the  writings  of  other  authors."  We  remember  that 
Meuc-echmus,  and  all  his  successors  down  to  Apollonius,  considered  only 
sections  of  right  cones  by  a  plane  perpendicular  to  their  sides,  and  that 
the  three  sections  were  obtained  each  from  a  different  cone.  Apol- 
lonius introduced  an  important  generaUsation.  He  produced  all  the 
sections  from  one  and  the  same  cone,  whether  right  or  scalene,  and 
by  sections  which  may  or  may  not  be  perpendicular  to  its  sides.  The 
old  names  for  the  three  curv^es  were  now  no  longer  applicable.  Instead 
of  calling  the  three  curves,  sections  of  the  "acute-angled,"  "right- 
an-^led,"  and  "obtuse-angled"  cone,  he  called  them  ellipse,  parabola, 
and  hyperbola,  respectively.  To  be  sure,  we  find  the  words  "parabola  " 
and  "ellipse"  in  the  works  of  Archimedes,  but  they  are  probably  only 
interpolations.  The  word  "ellipse"  was  applied  because  y-<px,  p 
being  the  parameter;  the  word  "parabola"  was  introduced  because 
y-  =  px,  and  the  term  "h}'perbola"  because  y'^>px. 

The  treatise  of  Apollonius  rests  on  a  unique  property  of  conic  sec- 
tions, which  is  derived  directly  from  the  nature  of  the  cone  in  which 
these  sections  are  found.  How  this  property  forms  the  key  to  the 
system  of  the  ancients  is  told  in  a  masterly  way  by  M.  Chasles.^ 
"Conceive,"  says  he,  "an  oblique  cone  on  a  circular  base;  the  straight 
line  drawTi  from  its  summit  to  the  centre  of  the  circle  forming  its  base 
is  called  the  axis  of  the  cone.  The  plane  passing  through  the  axis, 
perpendicular  to  its  base,  cuts  the  cone  along  two  lines  and  determines 
in  the  circle  a  diameter;  the  triangle  having  this  diameter  for  its  base 

'  n.  G.  Zeuthen,  Die  Lehre  vo7i  den  Kegehchnilten  im  Allerthum,  Kopenhagen, 
i886,  p.  502. 

2  M.  Chasles,  Geschichle  der  Geomclrie.  Aus  dem  Franzosischen  ubertragen  durch, 
Dr.  L.  A.  Sohncke,  Halle,  1839,  p.  15. 


40  A  HISTORY  OF  MATHEMATICS 

and  the  two  lines  for  its  sides,  is  called  the  triangle  through  the  axis. 
In  the  formation  of  his  conic  sections,  Apollonius  supposed  the  cutting 
plane  to  be  perpendicular  to  the  plane  of  the  triangle  through  the 
axis.  The  points  in  which  this  plane  meets  the  two  sides  of  this  tri- 
angle are  the  vertices  of  the  curve;  and  the  straight  line  which  joins 
these  two  points  is  a  diameter  of  it.  Apollonius  called  this  diameter 
latiis  transversum.  At  one  of  the  two  vertices  of  the  curve  erect  a  per- 
pendicular {latus  rectum)  to  the  plane  of  the  triangle  through  the 
axis,  of  a  certain  length,  to  be  determined  as  we  shall  specify  later, 
and  from  the  extremity  of  this  perpendicular  draw  a  straight  line  to 
the  other  vertex  of  the  curve;  now,  through  any  point  whatever  of 
the  diameter  of  the  curve,  draw  at  right  angles  an  ordinate:  the  square 
of  this  ordinate,  comprehended  between  the  diameter  and  the  curve, 
will  be  equal  to  the  rectangle  constructed  on  the  portion  of  the  ordinate 
comprised  between  the  diameter  and  the  straight  line,  and  the  part 
of  the  diameter  comprised  between  the  first  vertex  and  the  foot  of  the 
ordinate.  Such  is  the  characteristic  property  which  Apollonius  recog- 
nises in  his  conic  sections  and  which  he  uses  for  the  purpose  of  in- 
ferring from  it,  by  adroit  transformations  and  deductions,  nearly  all 
the  rest.  It  plays,  as  we  shall  see,  in  his  hands,  almost  the  same  role 
as  the  equation  of  the  second  degree  with  two  variables  (abscissa  and 
ordinate)  in  the  system  of  analytic  geometry  of  Descartes."  Apol- 
lonius made  use  of  co-ordinates  as  did  Menaechmus  before  him.^ 
Chasles  continues: 

"It  will  be  observed  from  this  that  the  diameter  of  the  curve  and 
the  perpendicular  erected  at  one  of  its  extremities  suffice  to  construct 
the  curve.  These  are  the  two  elements  which  the  ancients  used,  with 
which  to  establish  their  theory  of  conies.  The  perpendicular  in  ques- 
tion was  called  by  them  latus  erectum;  the  modems  changed  this  name 
first  to  that  of  latus  rectum,  and  afterwards  to  that  of  parameter.'" 

The  first  book  of  the  Conic  Sections  of  Apollonius  is  almost  wholly 
devoted  to  the  generation  of  the  three  principal  conic  sections. 

The  second  book  treats  mainly  of  asymptotes,  axes,  and  diameters. 

The  third  book  treats  of  the  equality  or  proportionality  of  triangles, 
rectangles,  or  squares,  of  which  the  component  parts  are  determined 
by  portions  of  transversals,  chords,  asymptotes,  or  tangents,  which 
are  frequently  subject  to  a  great  number  of  conditions.  It  also  touches 
the  subject  of  foci  of  the  ellipse  and  hyperbola. 

In  the  fourth  book,  Apollonius  discusses  the  harmonic  division  of 
straight  lines.  He  also  examines  a  system  of  two  conies,  and  shows 
that  they  cannot  cut  each  other  in  more  than  four  points.  He  inves- 
tigates the  various  possible  relative  positions  of  two  conies,  as,  for 
instance,  when  they  have  one  or  two  points  of  contact  with  each  other. 

The  fifth  book  reveals  better  than  any  other  the  giant  intellect  of 
its  author.  Difficult  questions  of  maxima  and  minima,  of  which  ^ew 
IT.  L.  Heath,  Apollonius  of  F^r^a,  Cambridge,  1S96,  p.  CXV. 


GREEK  GECMETRY  41 

examples  are  found  in  earlier  works,  are  here  treated  most  exhaustively. 
The  subject  investigated  is,  to  find  the  longest  and  shortest  lines  that 
can  be  drawn  from  a  given  point  to  a  conic.  Here  are  also  found  the 
germs  of  the  subject  of  evoliites  and  centres  of  osculation. 

The  sixth  book  is  on  the  similarity  of  conies. 

The  seventh  book  is  on  conjugate  diameters. 

The  eighth  book,  as  restored  by  Halley,  continues  the  subject  of 
conjugate  diameters. 

It  is  worthy  of  notice  that  Apollonius  nowhere  introduces  the 
notion  of  directrix  for  a  conic,  and  that,  though  he  incidentally  dis- 
covered t\vt  focus  of  an  ellipse  and  hyperbola,  he  did  not  discover  the 
focus  of  a  parabola.^  Conspicuous  in  his  geometry  is  also  the  absence 
of  technical  terms  and  symbols,  which  renders  the  proofs  long  and 
cumbrous.  R.  C.  Archibald  claims  that  Apollonius  was  familiar  with 
the  centres  of  similitude  of  circles,  usually  attributed  to  Monge. 
T.  L.  Heath"  comments  thus:  "The  principal  machinery  used  by 
Apollonius  as  well  as  by  the  earlier  geometers  comes  under  the  head 
of  what  has  been  not  inappropriately  called  a  geometrical  algebra.'^ 

The  discoveries  of  Archimedes  and  Apollonius,  says  M.  Chasles. 
marked  the  most  brilliant  epoch  of  ancient  geometry.  Two  questions 
which  have  occupied  geometers  of  all  periods  may  be  regarded  as 
having  originated  with  them.  The  first  of  these  is  the  quadrature  of 
curvilinear  figures,  which  gave  birth  to  the  infinitesimal  calculus.  The 
second  is  the  theory  of  conic  sections,  which  was  the  prelude  to  the 
theory  of  geometrical  curves  of  all  degrees,  and  to  that  portion  of 
geometry  which  considers  only  the  forms  and  situations  ot  figures 
and  uses  only  the  intersection  of  lines  and  surfaces  and  the  ratios  of 
rectilineal  distances.  These  two  great  divisions  of  geometry  may  be 
designated  by  the  names  of  Geometry  of  Measurements  and  Geometry 
of  Forms  and  Situations,  or,  Geometry  of  Archimedes  and  of  Apol- 
lonius. 

Besides  the  Conic  Sections,  Pappus  ascribes  to  Apollonius  the  fol- 
lowing works:  On  Contacts,  Plane  Loci,  Inclinations,  Section  of  an  Area, 
Determinate  Section,  and  gives  lemmas  from  which  attempts  have 
been  made  to  restore  the  lost  originals.  Two  books  on  De  Sectione 
Rationis  have  been  found  in  the  Arabic.  The  book  on  Contacts,  as 
restored  by  F.  Vieta,  contains  the  so-called  "Apollonian  Problem": 
Given  three  circles,  to  find  a  fourth  which  shall  touch  the  three. 

Euclid,  Archimedes,  and  Apollonius  brought  geometry  to  as  high 
a  state  of  perfection  as  it  perhaps  could  be  brought  without  first  in- 
troducing some  more  general  and  more  powerful  method  than  the  old 
method  of  exhaustion.  A  briefer  symbolism,  a  Cartesian  geometry, 
an  infinitesimal  calculus,  were  needed.     The  Greek  mind  was  not 

^  J.  Gow,  op.  cit.,  p.  252. 

-  T.  L.  Heath,  Apollonius  of  Perga,  edited  in  modern  notation.    Cambridge,  1896, 


42  A  HISTORY  OF  MATHEMATICS 

adapted  to  the  invention  of  general  methods.  Instead  of  a  cUmb  to 
still  loftier  heights  we  observe,  therefore,  on  the  part  of  later  Greek 
geometers,  a  descent,  during  which  they  paused  here  and  there  to  look 
around  for  details  which  had  been  passed  by  in  the  hast}-  ascent.^ 

Among  the  earliest  successors  of  Apollonius  was  Nicomedes.  Noth- 
ing definite  is  known  of  him,  except  that  he  invented  the  conchoid 
("mussel-like"),  a  curve  of  the  fourth  order.  He  devised  a  little 
machine  by  which  the  curve  could  be  easily  described.  With  aid  of 
the  conchoid  he  duplicated  the  cube.  The  curve  can  also  be  used  for 
trisecting  angles  in  a  manner  resembling  that  in  the  eighth  lemma  of 
Archimedes.  Proclus  ascribes  this  mode  of  trisection  to  Nicomedes, 
but  Pappus,  on  the  other  hand,  claims  it  as  his  own.  The  conchoid 
was  used  by  Newton  in  constructing  curves  of  the  third  degree. 

About  the  time  of  Nicomedes  (say,  i8o  b.  c),  flourished  also 
Diodes,  the  inventor  of  the  cissoid  ("ivy-like").  This  curve  he  used 
for  finding  two  mean  proportionals  between  two  given  straight  lines. 
The  Greeks  did  not  consider  the  companion-curve  to  the  cissoid; 
in  fact,  they  considered  only  the  part  of  the  cissoid  proper  which 
lies  inside  the  circle  used  in  constructing  the  curv^e.  The  part  of  the 
area  of  the  circle  left  over  when  the  two  circular  areas  on  the  concave 
sides  of  the  branches  of  the  curve  are  removed,  looks  somewhat  like 
an  ivy-leaf.  Hence,  probably,  the  name  of  the  curve.  That  the  two 
branches  extend  to  infinity  appears  to  have  been  noticed  first  by  G.  P. 
de  Roberal  in  1640  and  then  by  R.  de  Sluse.^ 

About  the  life  of  Perseus  we  know  as  little  as  about  that  of  Nico- 
medes and  Diodes.  He  lived  some  time  between  200  and  100  b.  c. 
From  Heron  and  Geminus  we  learn  that  he  wrote  a  work  on  the  spire, 
a  sort  of  anchor-ring  surface  described  by  Heron  as  being  produced  by 
the  revolution  of  a  circle  around  one  of  its  chords  as  an  axis.  The 
sections  of  this  surface  yield  peculiar  curves  called  spiral  sections, 
which,  according  to  Geminus,  were  thought  out  by  Perseus.  These 
curves  appear  to  be  the  same  as  the  Hippopede  of  Eudoxus. 

Probably  somewhat  later  than  Perseus  lived  Zenodorus.  He  wrote 
an  interesting  treatise  on  a  new  subject;  namely,  isoperiynetrical figures. 
Fourteen  propositions  are  preserved  by  Pappus  and  Theon.  Here 
are  a  few  of  them:  Of  isoperimetrical,  regular  polygons,  the  one  having 
the  largest  number  of  angles  has  the  greatest  area;  the  circle  has  a 
greater  area  than  any  regular  polygon  of  equal  periphery;  of  all  iso- 
perimentrical  polygons  of  n  sides,  the  regular  is  the  greatest;  of 
all  solids  having  surfaces  equal  in  area,  the  sphere  has  the  greatest 
volume. 

Hypsicles  (between  200  and  100  b.  c.)  was  supposed  to  be  the 
author  of  both  the  fourteenth  and  fifteenth  books  of  Euclid,  but  recent 
critics  are  of  opinion  that  the  fifteenth  book  was  written  by  an  author 

1  M.  Cantor,  op.  cit.,  Vol.  I,  3  Aufl.,  1907,  p^35o. 

"  G.  Loria,  Ebene  Ciirven,  transl.  by  F.  Schiitte,  I,  iqio,  p,  37. 


GREEK  GEOMETRY  43 

will)  lived  several  centuries  after  Christ.  The  fourteenth  book  con- 
tains seven  elegant  theorems  on  regular  solids.  A  treatise  of  Hypsicles 
on  Risings  is  of  interest  because  it  gives  the  division  of  the  circum- 
ference into  360  degrees  after  the  fashion  of  the  Babylonians. 

Hipparchus  of  Nica^a  in  Bithynia  was  the  greatest  astronomer  of 
antiquity.  He  took  astronomical  observations  between  161  and  127 
B.  c.  He  established  inductively  the  famous  theory  of  epicycles  and 
eccentrics.  As  might  be  expected,  he  was  interested  in  mathematics, 
not  per  se,  but  only  as  an  aid  to  astronomical  inquiry.  No  mathe- 
matical writings  of  his  are  extant,  but  Theon  of  Alexandria  informs  us 
that  Hipparchus  originated  the  science  of  trigonometry,  and  that  he 
calculated  a  "table  of  chords"  in  twelve  books.  Such  calculations 
must  have  required  a  ready  knowledge  of  arithmetical  and  algebraical 
operations.  He  possessed  arithmetical  and  also  graphical  devices  for 
solving  geometrical  problems  in  a  plane  and  on  a  sphere.  He  gives 
indication  of  having  seized  the  idea  of  co-ordinate  representation,  found 
earlier  in  Apollonius. 

About  100  B.  c.  flourished  Heron  the  Elder  of  Alexandria.  He  was 
the  pupil  of  Ctesibius,  who  was  celebrated  for  his  ingenious  mechanical 
inventions,  such  as  the  hydraulic  organ,  the  water-clock,  and  catapult. 
It  is  believed  by  some  that  Heron  was  a  son  of  Ctesibius.  He  ex- 
hibited talent  of  the  same  order  as  did  his  master  by  the  invention  of 
the  eolipile  and  a  curious  mechanism  known  as  ''Heron's  fountain." 
Great  uncertainty  exists  concerning  his  writings.  Most  authorities 
believe  him  to  be  the  author  of  an  important  Treatise  on  the  Dioptra, 
of  which  there  exist  three  manuscript  copies,  quite  dissimilar.  But 
M.  Marie  ^  thinks  that  the  Dioptra  is  the  Avork  of  Heron  the  Younger, 
who  lived  in  the  seventh  or  eighth  century  after  Christ,  and  that 
Geodesy,  another  book  supposed  to  be  by  Heron,  is  only  a  corrupt  and 
defectiv^e  copy  of  the  former  work.  Dioptra  contains  the  important 
formula  for  finding  the  prea  of  a  triangle  expressed  in  terms  of  its 
sides;  its  derivation  is  quite  laborious  and  yet  exceedingly  ingenious. 
"It  seems  to  me  difficult  to  believe,"  says  Chasles,  "that  so  beautiful 
a  theorem  should  be  found  in  a  work  so  ancient  as  that  of  Heron  the 
Elder,  without  that  some  Greek  geometer  should  have  thought  to 
cite  it."  Marie  lays  great  stress  on  this  silence  of  the  ancient  writers, 
and  argues  from  it  that  the  true  author  must  be  Heron  the  Younger 
or  some  writer  much  more  recent  than  Heron  the  Elder.  But  no  re- 
hable  evidence  has  been  found  that  there  actually  existed  a  second 
mathematician  by  the  name  of  Heron.  P.  Tannery  has  shown  that, 
in  applying  this  formula,  Heron  found  the  irrational  square  roots  by 

the  approximation,  ■\/yl~i(a+— ),  where  a-  is  the  square  nearest  to 

'  Maximilien  Marie,  Ilhloire  dcs  sciences  mathemaliques  cl  physiques.  Paris, 
Tome  I,  1883,  p.  T78. 


44  A  HISTORY  OF  MATHEMATICS 

A.    When  a  more  accurate  value  was  wanted,  Heron  took  ^(a-\ — ) 

in  the  place  of  a  in  the  above  formula.  Apparently,  Heron  some- 
times found  square  and  cube  roots  also  by  the  method  of  "double 
false  position." 

"Dioptra,"  says  Venturi,  were  instruments  which  had  great  re- 
semblance to  our  modern  theodolites.  The  book  Dioptra  is  a  treatise 
on  geodesy  containing  solutions,  with  aid  of  these  instruments,  of  a 
large  number  of  questions  in  geometry,  such  as  to  find  the  distance 
between  two  points,  of  which  one  only  is  accessible,  or  between  two 
points  which  are  visible  but  both  inaccessible;  from  a  given  point  to 
draw  a  perpendicular  to  a  line  which  cannot  be  approached;  to  find 
the  difference  of  level  between  two  points;  to  measure  the  area  of  a 
field  without  entering  it. 

Heron  was  a  practical  surveyor.  This  may  account  for  the  fact 
that  his  writings  bear  so  little  resemblance  to  those  of  the  Greek 
authors,  who  considered  it  degrading  the  science  to  apply  geometry  to 
surveying.  The  character  of  his  geometry  is  not  Grecian,  but  de- 
cidedly Egyptian.  This  fact  is  the  more  surprising  when  we  consider 
that  Heron  demonstrated  his  familiarity  with  Euclid  by  writing  a  com.- 
mentary  on  the  Elements.  Some  of  Heron's  formulas  point  to  an  old 
Egyptian  origin.    Thus,  besides  the  above  exact  formula  for  the  area 

of  a  triangle  in  terms  of  its  sides.  Heron  gives  the  formula  • — ^ — -"X  -, 

which  bears  a  striking  likeness  to  the  formula  ^X '  f^r 

finding  the  area  of  a  quadrangle,  found  in  the  Edfu  inscriptions. 
There  are,  moreover,  points  of  resemblance  between  Heron's  writings 
and  the  ancient  Ahmes  papyrus.  Thus  Ahmes  used  unit-fractions 
exclusively  (except  the  fraction  f) ;  Heron  uses  them  of tener  than  other 
fractions.  Like  Ahmes  and  the  priests  at  Edfu,  Heron  divides  com- 
plicated figures  into  simpler  ones  by  drawing  auxiliary  lines;  like  them, 
he  shows,  throughout,  a  special  fondness  for  the  isosceles  trapezoid. 

The  writings  of  Heron  satisfied  a  practical  want,  and  for  that  reason 
were  borrowed  extensively  by  other  peoples.  We  find  traces  of  them 
in  Rome,  in  the  Occident  during  the  Middle  Ages,  and  even  in  India. 

The  works  attributed  to  Heron,  including  the  newly  discovered 
Metrica  pubUshed  in  1903,  have  been  edited  by  J.  H.  Heiberg, 
H.  Schone  and  W.  Schmidt. 

Geminus  of  Rhodes  (about  70  b.  c.)  pubHshed  an  astronomical  work 
still  extant.  He  wrote  also  a  book,  now  lost,  on  the  Arrangement  oj 
Mathematics,  which  contained  many  valuable  notices  of  the  early 
history  of  Greek  mathematics.  Proclus  and  Eutocius  quote  it  fre- 
quently.   Theodosius  is  the  author  of  a  book  of  little  merit  on  the 


GREEK  GEOMETRY  45 

geometry  of  the  sphere.  Investigations  due  to  P.  Tannery  and  A.  A. 
Bjornbo  ^  seem  to  indicate  that  the  mathematician  Theodosius  was 
not  Theodosius  of  Tripolis,  as  formerly  supposed,  but  was  a  resident 
of  Bithynia  and  contemporary  of  Hipparchus.  Dionysodoms  of 
Amisus  in  Pontus  applied  the  intersection  of  a  parabola  and  hyperbola 
to  the  solution  of  a  problem  which  Archimedes,  in  his  Sphere  atid 
Cylinder,  had  left  incomplete.  The  problem  is  "to  cut  a  sphere  so 
that  its  segments  shall  be  in  a  given  ratio." 

We  have  now  sketched  the  progress  of  geometry  down  to  the  time 
of  Christ.  Unfortunately,  very  little  is  known  of  the  history  of  geom- 
etry between  the  time  of  Apollonius  and  the  beginning  of  the  Christian 
era.  The  names  of  quite  a  number  of  geometers  have  been  mentioned, 
but  very  few  of  their  works  are  now  extant.  It  is  certain,  however, 
that  there  were  no  mathematicians  of  real  genius  from  Apollonius  to 
Ptolemy,  excepting  Hipparchus  and  perhaps  Heron. 

The  Second  Alexandrian  School 

The  close  of  the  dynasty  of  the  Lagides  which  ruled  Egypt  from  the 
time  of  Ptolemy  Soter,  the  builder  of  Alexandria,  for  300  years;  the 
absorption  of  Egypt  into  the  Roman  Empire;  the  closer  commercial 
relations  between  peoples  of  the  East  and  of  the  West;  the  gradual 
decline  of  paganism  and  spread  of  Christianity, — these  events  were 
of  far-reaching  influence  on  the  progress  of  the  sciences,  which  then 
had  their  home  in  Alexandria.  Alexandria  became  a  commercial  and 
intellectual  emporium.  Traders  of  all  nations  met  in  her  busy  streets, 
and  in  her  magnificent  Library,  museums,  lecture-halls,  scholars  from 
the  East  mingled  with  those  of  the  West;  Greeks  began  to  study  older 
literatures  and  to  compare  them  with  their  own.  In  consequence  of 
this  interchange  of  ideas  the  Greek  philosophy  became  fused  with 
Oriental  philosophy.  Neo-Pythagoreanism  and  Neo-Platonism  were 
the  names  of  the  modified  systems.  These  stood,  for  a  time,  in  op- 
position to  Christianity.  The  study  of  Platonism  and  Pythagorean 
mysticism  led  to  the  re\'ival  of  the  theory  of  numbers.  Perhaps  the 
dispersion  of  the  Jews  and  their  introduction  to  Greek  learning  helped 
in  bringing  about  this  revival.  The  theory  of  numbers  became  a 
favorite  study.  This  new  line  of  mathematical  inquiry  ushered  in 
what  we  may  call  a  new  school.  There  is  no  doubt  that  even  now 
geometry  continued  to  be  one  of  the  most  important  studies  in  the 
Alexandrian  course.  This  Second  Alexandrian  School  may  be  said  to 
begin  with  the  Christian  era.  It  was  made  famous  by  the  names  of 
Claudius  Ptolemaeus,  Diophantus,  Pappus,  Theon  of  Smyrna,  Theon 
of  Alexandria,  lamblichus.  Porphyrins,  and  others. 

By  the  side  of  these  we  may  place  Serenus  of  Antina:ia,  as  ha\-ing 

'Axel  Anthon  RjOrnbo  (1874-1911)  of  Copenhagen  was  a  historian  of  mathe- 
matics.   See  Bibliothcca  maihemalica,  3  S.,  ^'  J.  12,  1911-12,  pp.  337-344- 


46  A  HISTORY  OF  MATHEMATICS 

been  connected  more  or  less  with  this  new  school.  He  wrote  on  sec- 
tions of  the  cone  and  cylinder,  in  two  books,  one  of  which  treated 
only  of  the  triangular  section  of  the  cone  through  the  apex.  He  solved 
the  problem,  ''given  a  cone  (cylinder),  to  find  a  cylinder  (cone),  so 
that  the  section  of  both  by  the  same  plane  gives  similar  ellipses."  Of 
particular  interest  is  the  following  theorem,  which  is  the  foundation 
of  the  modem  theory  of  harmonics:  If  from  D  we  draw  DF,  cutting 
the  triangle  ABC,  and  choose  H  on  it,  so  that  DE:DF  =  EH  :  HF, 
and  if  we  draw  the  line  AH,  then  every  transversal  through  D,  such 
as  DG,  will  be  divided  by  yl^  so  that  DK  :DG=KJ  :  JG.  Menelaus 
of  Alexandria  (about  98  a.  d.)  was  the  author  of  Sphcerica,  a  work 
extant  in  Hebrew  and  Arabic,  but  not  in  Greek.    In  it  he  proves  the 

theorems  on  the  congruence  of 
spherical  triangles,  and  describes 
their  properties  in  much  the  same 
way  as  Euclid  treats  plane  tri- 
angles. In  it  are  also  found  the 
theorems  that  the  sum  of  the  three 
sides  of  a  spherical  triangle  is  less 
than  a  great  circle,  and  that  the 
sura  of  the  three  angles  exceeds  two  right  angles.  Celebrated  are  two 
theorems  of  his  on  plane  and  spherical  triangles.  The  one  on  plane  tri- 
angles is  that,  "  if  the  three  sides  be  cut  by  a  straight  line,  the  product  of 
the  three  segments  which  have  no  common  extremity  is  equal  to  the 
product  of  the  other  three."  L.  N.  M.  Carnot  makes  this  proposition, 
known  as  the  "lemma  of  Menelaus,"  the  base  of  his  theory  of  trans- 
versals. The  corresponding  theorem  for  spherical  triangles,  the  so- 
called  "regula  sex  quantitatum,"  is  obtained  from  the  above  by 
reading  "chords  of  three  segments  doubled,"  in  place  of  "three  seg- 
ments." 

Claudius  Ptolemy,  a  celebrated  astronomer,  was  a  native  of  Egypt. 
Nothing  is  known  of  his  personal  history  except  that  he  flourished  in 
Alexandria  in  139  a.  d.  and  that  he  made  the  earliest  astronomical 
observations  recorded  in  his  works,  in  125  a.  d.,  the  latest  in  151  a.  d. 
The  chief  of  his  works  are  the  Syntaxis  Mathematica  (or  the  Almagest, 
as  the  Arabs  call  it)  and  the  Geographica,  both  of  which  are  extant. 
The  former  work  is  based  partly  on  his  own  researches,  but  mainly 
on  those  of  Hipparchus.  Ptolemy  seems  to  have  been  not  so  much  of 
an  independent  investigator,  as  a  corrector  and  improver  of  the  work 
of  his  great  predecessors.  The  Almagest  1  forms  the  foundation  of 
all  astronomical  science  down  to  N.  Copernicus.  The  fundamental 
idea  of  his  system,  the  "Ptolemaic  System,"  is  that  the  earth  is  in  the 
centre  of  the  universe,  and  that  the  sun  and  planets  revolve  around 
the  earth.    Ptolemy  did  considerable  for  mathematics.    He  created, 

*  On  the  importance  of  the  Almagest  in  the  history  of  astronomy,  consult  P. 
Tannery,  Recherches  sur  Vhistoirc  de  I'aslronomie,  Paris,  1893. 


GREEK  GEOMETRY  47 

for  astronomical  use,  a  trigonometry  remarkably  perfect  in  form.    The 
foundation  of  this  science  was  laid  by  the  illustrious  Hipparchus. 

The  Almagest  is  in  13  books.  Chapter  9  of  the  first  book  shows  how 
to  calculate  tables  of  chords.  The  circle  is  divided  into  360  degrees, 
each  of  which  is  halved.  The  diameter  is  divided  into  120  divisions; 
each  of  these  into  60  parts,  which  are  again  subdivided  into  60  smaller 
parts.  In  Latin,  these  parts  were  called  partes  mimitcB  primce  and 
partes  miniitcB  secuiidcr.  Hence  our  names,  "minutes"  and  "seconds." 
The  sexagesimal  method  of  dividing  the  circle  is  of  Babylonian  origin, 
and  was  known  to  Geminus  and  Hipparchus.  But  Ptolemy's  method 
of  calculating  chords  seems  original  with  him.  He  first  proved  the 
proposition,  now  appended  to  Euclid  VI  (D),  that  "the  rectangle 
contained  by  the  diagonals  of  a  quadrilateral  figure  inscribed  in  a 
circle  is  equal  to  both  the  rectangles  contained  by  its  opposite  sides." 
He  then  shows  how  to  find  from  the  chords  of  two  arcs  the  chords  of 
their  sum  and  difference,  and  from  the  chord  of  any  arc  that  of  its 
half.  These  theorems  he  applied  to  the  calculation  of  his  tables  of 
chords.  The  proofs  of  these  theorems  are  very  pretty.  Ptolemy's 
construction  of  sides  of  a  regular  inscribed  pentagon  and  decagon  was 
given  later  by  C.  Clavius  and  L.  Mascheroni,  and  now  is  used  much 
by  engineers.  Let  the  radius  BD  be  _L  to  ^C, 
DE^EC.  Make  EF^EB,  then  BF  is  the  side  of 
the  pentagon  and  DF  is  the  side  of  the  decagon. 

Another  chapter  of  the  first  book  in  the  Alma- 
gest is  devoted  to  trigonometry,  and  to  spherical 
trigonometry  in  particular.  Ptolemy  proved  the 
"lemma  of  Menelaus,"  and  also  the  "regula  sex  quantitatum." 
Upon  these  propositions  he  built  up  his  trigonometry.  In  trigono- 
metric computations,  the  Greeks  did  not  use,  ^s  did  the  Hindus,  half 
the  chord  of  twice  the  arc  (the  "sine");  the  Greeks  used  instead 
the  whole  chord  of  double  the  arc.  Only  in  graphic  constructions, 
referred  to  again  later,  did  Ptolemy  and  his  predecessors  use  half  the 
chord  of  double  the  arc.  The  fundamental  theorem  of  plane  trigo- 
nometry, that  two  sides  of  a  triangle  are  to  each  other  as  the  chords 
of  double  the  arcs  measuring  the  angles  opposite  the  two  sides,  was 
not  stated  ex-^^ylicitly  by  Ptolemy,  but  was  contained  implicitly  in  other 
theorems.  More  complete  are  the  propositions  in  spherical  trigo- 
nometry. 

The  fact  that  trigonometry  was  cultivated  not  for  its  own  sake,  but 
to  aid  astronomical  inquiry,  explains  the  rather  startling  fact  that 
spherical  trigonometry  came  to  exist  in  a  developed  state  earlier  than 
plane  trigonometry. 

The  remaining  books  of  the  Almagest  are  on  astronomy.  Ptolemy 
has  written  other  works  which  have  little  or  no  bearing  on  mathe- 
matics, except  one  on  geometry.  Extracts  from  this  book,  made  by 
Proclus,  indicate  that  Ptolemy  did  not  regard  the  parallel-axiom  of 


48  A  HISTORY  OF  MATHEMATICS 

Euclid  as  self-evident,  and  that  Ptolemy  was  the  first  of  the  long  line 
of  geometers  from  ancient  time  down  to  our  own  who  toiled  in  the  vain 
attempt  to  prove  it.  The  untenable  part  of  his  demonstration  is  the 
assertion  that,  in  case  of  parallelism,  the  sum  of  the  interior  angles  on 
one  side  of  a  transversal  must  be  the  same  as  their  sum  on  the  other 
side  of  the  transversal.  Before  Ptolemy  an  attempt  to  improve  the 
theory  of  parallels  was  made  by  Posidonius  (first  cent.  b.  c.)  who  de- 
fined parallel  lines  as  lines  that  are  coplanar  and  equidistant.  From 
an  Arabic  writer,  Al-Nirizi  (ninth  cent.)  it  appears  that  Simplicius 
brought  forward  a  proof  of  the  5  th  postulate,  based  upon  this  def- 
inition, and  due  to  his  friend  Aganis  (Geminus?).^ 

In  the  making  of  maps  of  the  earth's  surface  and  of  the  celestial 
sphere,  Ptolemy  (following  Hipparchus)  used  stereographic  projection. 
The  eye  is  imagined  to  be  at  one  of  the  poles,  the  projection  being 
thrown  upon  the  equatorial  plane.  He  devised  an  instrument,  a  form 
of  astrolabe  planisphere,  which  is  a  stereographic  projection  of  the 
celestial  sphere. 2  Ptolemy  wrote  a  monograph  on  the  analemma  which 
was  a  figure  involving  orthographic  projections  of  the  celestial  sphere 
upon  three  mutually  perpendicular  planes  (the  horizontal,  meridian 
and  vertical  circles) .  The  analemma  was  used  in  determining  positions 
of  the  sun,  the  rising  and  setting  of  the  stars.  The  procedure  was 
probably  known  to  Hipparchus  and  the  older  astronomers.  It  fur- 
nished a  graphic  method  for  the  solution  of  spherical  triangles  and  was 
used  subsequently  by  the  Hindus,  the  Arabs,  and  Europeans  as  late 
as  the  seventeenth  century.^ 

Two  prominent  mathematicians  of  this  time  were  Nicomachus  and 
Theon  of  Smyrna.  Their  favorite  study  was  theory  of  numbers. 
The  investigations  in  this  science  culminated  later  in  the  algebra  of 
Diophantus.  But  no  important  geometer  appeared  after  Ptolemy 
for  150  years.  An  occupant  of  this  long  gap  was  Sextus  Julius 
Africanus,  who  wrote  an  unimportant  work  on  geometry  applied 
to  the  art  of  war,  entitled  Cestes.  Another  was  the  sceptic,  Sextus 
Empiricus  (200  a.  d.);  he  endeavored  to  elucidate  Zeno's  "Arrow" 
by  stating  another  argument  equally  paradoxical  and  therefore  far 
from  illuminating:  Men  never  die,  for  if  a  man  die,  it  must  either 
be  at  a  time  when  he  is  alive,  or  at  a  time  when  he  is  not  alive; 
hence  he  never  dies.  Sextus  Empiricus  advanced  also  the  paradox, 
that,  when  a  line  rotating  in  a  plane  about  one  of  its  ends  describes 
a  circle  with  each  of  its  points,  these  concentric  circles  are  of  un- 
equal area,  yet  each  circle  must  be  equal  to  the  neighbouring  circle 
which  it  touches.^ 

iR.  Bonola,  Non-Eiiclidean  Geometry,  trans,  by  H.  S.  Carslaw,  Chicago,  191 2, 
pp.  3-8.    Robert  Bonola  (1875-1911)  was  professor  in  Rome. 

'See  M.  Latham,  "  The  Astrolabe,"  A)n.  Math.  Monthly,  Vol.  24,  1917,  p.  162. 

^  See  A.  V.  Braunmtihl,  Geschichte  dcr  Trigonomdrie,  Leipzig,  I,  1900,  p.  11. 
Alexander  von  Braunmiihl  (1853-1908)  was  professor  at  the  technical  high  school 
in  Munich. 


GREEK  GEOMETRY  49 

Pappus,  probably  born  about  340  a.  d.,  in  Alexandria,  was  the 
■ast  great  mathematician  of  the  Alexandrian  school.  His  genius  was 
inferior  to  that  of  Archimedes,  Apollonius,  and  Euclid,  who  flourished 
over  500  years  earlier.  But  living,  as  he  did,  at  a  period  when  interest 
in  geometry  was  declining,  he  towered  above  his  contemporaries  "like 
the  peak  of  Teneriffa  above  the  Atlantic."  He  is  the  author  of  a 
Commentary  on  i/ie  Almagest,  a  Commentary  on  Euclid's  Elements,  a 
Commoitary  on  the  Analemma  of  Diodorus, — a  writer  of  whom  nothing 
is  known.  All  these  works  are  lost.  Proclus,  probably  ciuoting  from 
the  Commentary  on  Euclid,  says  that  Pappus  objected  to  the  state- 
ment that  an  angle  equal  to  a  right  angle  is  always  itself  a  right 
angle. 

The  only  work  of  Pappus  still  extant  is  his  Mathematical  Collections. 
This  was  originally  in  eight  books,  but  the  first  and  portions  of  the 
second  are  now  missing.  The  Mathematical  Collections  seems  to  have 
been  \\Titten  by  Pappus  to  supply  the  geometers  of  his  time  with  a 
succinct  analysis  of  the  most  difficult  mathematical  works  and  to 
facilitate  the  study  of  them  by  explanatory  lemmas.  But  these 
lemmas  are  selected  very  freely,  and  frequently  have  little  or  no  con- 
nection with  the  subject  on  hand.  However,  he  gives  very  accurate 
summaries  of  the  works  of  which  he  treats.  The  Mathematical  Col- 
lections is  invaluable  to  us  on  account  of  the  rich  information  it  gives 
on  various  treatises  by  the  foremost  Greek  mathematicians,  which 
are  now  lost.  Mathematicians  of  the  last  century  considered  it  pos- 
sible to  restore  lost  work  from  the  resume  by  Pappus  alone. 

We  shall  now  cite  the  more  important  of  those  theorems  in  the 
Mathematical  Collections  which  are  supposed  to  be  original  with 
Pappus.  First  of  all  ranks  the  elegant  theorem  re-discovered  by  P. 
Giddin,  over  1000  years  later,  that  the  volume  generated  by  the 
revolution  of  a  plane  curve  which  lies  wholly  on  one  side  of  the  axis, 
equals  the  area  of  the  curve  multiplied  by  the  circumference  de- 
scribed by  its  center  of  gravity.  Pappus  proved  also  that  the  centre 
of  gravity  of  a  triangle  is  that  of  another  triangle  whose  vertices  lie 
upon  the  sides  of  the  first  and  divide  its  three  sides  in  the  same  ratio. 
In  the  fourth  book  are  new  and  brilliant  propositions  on  the  quadra- 
trix  which  indicate  an  intimate  acquaintance  with  curved  surfaces. 
He  generates  the  quadratrix  as  follows:  Let  a  spiral  line  be  drawn 
upon  a  right  circular  cylinder;  then  the  perpendiculars  to  the  axis 
of  the  cylinder  drawn  from  each  point  of  the  spiral  line  form  the 
surface  of  a  screw.  A  plane  passed  through  one  of  these  perpendicu- 
lars, making  any  convenient  angle  with  the  base  of  the  cylinder,  cuts 
the  screw-surface  in  a  curve,  the  orthogonal  projection  of  which  uj^on 
the  base  is  the  quadratrix.  A  second  mode  of  generation  is  no  less 
admirable:  If  we  make  the  spiral  of  Archimedes  the  base  of  a  right 

1  See  K.  Lasswitz,  Geschichie  der  Atomistik,  I,  Hamburg   unci  Leipzig,   1890, 
V-  148. 


50  A  HISTORY  OF  MATHEMATICS 

cylinder,  and  imagine  a  cone  of  revolution  having  for  its  axis  the  side 
of  the  cylinder  passing  through  the  initial  point  of  tie  spiral,  then 
this  cone  cuts  the  cylinder  in  a  curve  of  double  curvature.  The  per- 
pendiculars to  the  axis  drawn  through  every  point  in  this  curve  form 
the  surface  of  a  screw  which  Pappus  here  calls  the  plectoidal  surface. 
A  plane  passed  through  one  of  the  perpendiculars  at  any  convenient 
angle  cuts  that  surface  in  a  curve  whose  orthogonal  projection  upon 
the  plane  of  the  spiral  is  the  required  qiiadratrix.  Pappus  considers 
curves  of  double  curvature  still  further.  He  produces  a  spherical 
spiral  by  a  point  moving  uniformly  along  the  circumference  of  a 
great  circle  of  a  sphere,  while  the  great  circle  itself  revolves  uniformly 
around  its  diameter.  He  then  finds  the  area  of  that  portion  of  the 
surface  of  the  sphere  determined  by  the  spherical  spiral,  "a  complana- 
tion  which  claims  the  more  lively  admiration,  if  we  consider  that, 
although  the  entire  surface  of  the  sphere  was  known  since  Archimedes' 
time,  to  measure  portions  thereof,  such  as  spherical  triangles,  was 
then  and  for  a  long  time  afterwards  an  unsolved  problem."  ^  A 
question  which  was  brought  into  prominence  by  Descartes  and  Newton 
is  the  "problem  of  Pappus."  Given  several  straight  lines  in  a  plane, 
to  find  the  locus  of  a  point  such  that  when  perpendiculars  (or,  more 
generally,  straight  lines  at  given  angles)  are  drawn  from  it  to  the 
given  lines,  the  product  of  certain  ones  of  them  shall  be  in  a  given 
ratio  to  the  product  of  the  remaining  ones.  It  is  worth  noticing  that 
it  was  Pappus  who  first  found  the  focus  of  the  parabola  and  pro- 
pounded the  theory  of  the  involution  of  points.  He  used  the  directrix 
and  was  the  first  to  put  in  definite  form  the  definition  of  the  conic 
sections  as  loci  of  those  points  whose  distances  from  a  fixed  point 
and  from  a  fixed  line  are  in  a  constant  ratio.  He  solved  the  problem 
to  draw  through  three  points  lying  in  the  same  straight  line,  three 
straight  lines  which  shall  form  a  triangle  inscribed  in  a  given  circle. 
From  the  Mathematical  Collections  many  more  equally  difficult  the- 
orems might  be  quoted  which  are  original  with  Pappus  as  far  as  we 
know.  It  ought  to  be  remarked,  however,  that  he  has  been  charged 
in  three  instances  with  copying  theorems  without  giving  due  credit, 
and  that  he  may  have  done  the  same  thing  in  other  cases  in  which 
we  have  no  data  by  which  to  ascertain  the  real  discoverer.^ 

About  the  time  of  Pappus  lived  Theon  of  Alexandria.  He  brought 
out  an  edition  of  Euclid's  Elements  with  notes,  which  he  probably 
used  as  a  text-book  in  his  classes.  His  commentary  on  the  Almagest 
is  valuable  for  the  many  historical  notices,  and  especially  for  the 
specimens  of  Greek  ai'ithmetic  which  it  contains.  Theon's  daughter 
Hypatia,  a  woman  celebrated  for  her  beauty  and  modesty,  was  the 
last  Alexandrian  teacher  of  reputation,  and  is  said  to  have  been  an 

'  M.  Cantor,  op.  ciL,  Vol.  T,  3  Aufl..  1907,  p.  451. 

^  For  a  defence  of  Pappus  against  these  charges,  see  J.  H.  Weaver  in  Bull.  Am 
Muth.  Soc-,  Val   2.?,  1916,  pp.  131-133. 


GREEK  GEOMETRY  51 

abler  philosopher  and  mathematician  than  her  father.  Her  notes  on 
the  works  of  Diophantus  and  Apollonius  have  been  lost.  Her  tragic 
death  in  415  A.  d.  is  vividly  described  in  Kingsley's  Hypatia. 

From  now  on,  mathematics  ceased  to  be  cultivated  in  Alexandria. 
The  leading  subject  of  men's  thoughts  was  Christian  theology. 
Paganism  disappeared,  and  with  it  pagan  learning.  The  Neo-Platonic 
school  at  Athens  struggled  on  a  century  longer.  Proclus,  Isidorus,  and 
others  kept  up  the  "golden  chain  of  Platonic  succession."  Proclus, 
the  successor  of  Syrianus,  at  the  Athenian  school,  wrote  a  commentary 
on  Euclid's  Elements.  We  possess  only  that  on  the  first  book,  which 
is  valuable  for  the  information  it  contains  on  the  history  of  geometr}-. 
Damascius  of  Damascus,  the  pupil  of  Isidorus,  is  now  believed  to  be 
the  author  of  the  fifteenth  book  of  Euclid.  Another  pupil  of  Isidorus 
was  Eutocius  of  Ascalon,  the  commentator  of  Apollonius  and  Archi- 
medes. Simplicius  wrote  a  commentary  on  Aristotle's  De  Ccelo. 
Simplicius  reports  Zeno  as  saying:  "That  which,  being  added  to 
another,  does  not  make  it  greater,  and  being  taken  away  from  another 
does  not  make  it  less,  is  nothing."  According  to  this,  the  denial  of 
the  existence  of  the  infinitesimal  goes  back  to  Zeno.  This  momentous 
question  presented  itself  centuries  later  to  Leibniz,  who  gave  different 
answers.  The  report  made  by  Simplicius  of  the  quadratures  of  Anti- 
phon  and  Hippocrates  of  Chios  is  one  of  the  best  sources  of  historical 
information  on  this  point.^  In  the  year  529,  Justinian,  disapproving 
heathen  learning,  filially  closed  by  imperial  edict  the  schools  at 
Athens. 

As  a  rule,  the  geometers  of  the  last  500  years  showed  a  lack  of 
creative  power.    They  were  commentators  rather  than  discoverers. 

The  principal  characteristics  of  ancient  geometry  are: — 

(i)  A  wonderful  clearness  and  definiteness  of  its  concepts  and  an 
almost  perfect  logical  rigor  of  its  conclusions. 

(2)  A  complete  want  of  general  principles  and  methods.  Ancient 
geometry  is  decidedly  special.  Thus  the  Greeks  possessed  no  general 
method  of  drawing  tangents.  "The  determination  of  the  tangents 
to  the  three  conic  sections  did  not  furnish  any  rational  assistance  for 
drawing  the  tangent  to  any  other  new  cur\'e,  such  as  the  conchoid, 
the  cissoid,  etc."  In  the  demonstration  of  a  theorem,  there  were,  for 
the  ancient  geometers,  as  many  different  cases  requiring  separate 
])roof  as  there  were  different  positions  for  the  lines.  The  greatest 
geometers  considered  it  necessary  to  treat  all  possible  cases  inde- 
pendently of  each  other,  and  to  prove  each  with  equal  fulness.  To 
devise  methods  by  which  the  various  cases  could  all  be  disposed  of 
by  one  stroke,  Avas  beyond  the  power  of  the  ancients.  "If  we  com- 
pare a  mathematical  problem  with  a  huge  rock,  into  the  interior  of 
which  we  desire  to  penetrate,  then  the  work  of  the  Greek  mathe- 

*  See  F.  Rudio  in  Bibliotheca  mathemalica,  3  S.,  Vol.  3,  1902,  pp.  7-62. 


52  A  HISTORY  OF  MATHEMATICS 

maticians  appears  to  us  like  that  of  a  vigorous  stonecutter  who,  with 
chisel  and  hammer,  begins  with  indefatigable  perseverance,  from  with- 
out, to  crumble  the  rock  slowly  into  fragments;  the  modern  mathe- 
matician appears  like  an  excellent  miner,  who  first  bores  through  the 
rock  some  few  passages,  from  which  he  then  bursts  it  into  pieces 
with  one  powerful  blast,  and  brings  to  light  the  treasures  within."  ^ 


Greek  Arithmetic  and  Algebra 

Greek  mathematicians  were  in  the  habit  of  discriminating  between 
the  science  of  numbers  and  the  art  of  calculation.  The  former  they 
called  arithmetica,  the  latter  logistica.  The  drawing  of  this  distinction 
between  the  two  was  very  natural  and  proper.  The  difference  be- 
tween them  is  as  marked  as  that  between  theory  and  practice.  Among 
the  Sophists  the  art  of  calculation  was  a  favorite  study.  Plato,  on 
the  other  hand,  gave  considerable  attention  to  philosophical  arith- 
metic, but  pronounced  calculation  a  vulgar  and  childish  art. 

In  sketching  the  history  of  Greek  calculation,  we  shall  first  give  a 
brief  account  of  the  Greek  mode  of  counting  and  of  writing  numbers. 
Like  the  Egyptians  and  Eastern  nations,  the  earliest  Greeks  counted 
on  their  fingers  or  with  pebbles.  In  case  of  large  numbers,  the  pebbles 
were  probably  arranged  in  parallel  vertical  lines.  Pebbles  on  the 
first  line  represented  units,  those  on  the  second  tens,  those  on  the 
third  hundreds,  and  so  on.  Later,  frames  came  into  use,  in  which 
strings  or  wires  took  the  place  of  lines.  According  to  tradition, 
Pythagoras,  who  travelled  in  Egypt  and,  perhaps,  in  India,  first 
introduced  this  valuable  instrument  into  Greece.  The  abacus,  as  it 
is  called,  existed  among  different  peoples  and  at  different  times,  in 
various  stages  of  perfection.  An  abacus  is  still  employed  by  the 
Chinese  under  the  name  of  Swan-pan.  We  possess  no  specific  informa- 
tion as  to  how  the  Greek  abacus  looked  or  how  it  was  used.  Boethius 
says  that  the  Pythagoreans  used  with  the  abacus  certain  nine  signs 
called  apices,  which  resembled  in  form  the  nine  "Arabic  numerals." 
But  the  correctness  of  this  assertion  is  subject  to  grave  doubts. 

The  oldest  Grecian  numerical  symbols  were  the  so-called  Herodianic 
signs  (after  Herodianus,  a  Byzantine  grammarian  of  about  200  a.  d., 
who  describes  them).  These  signs  occur  frequently  in  Athenian  in- 
scriptions and  are,  on  that  account,  now  generally  called  Attic.  For 
some  unknown  reason  these  symbols  were  afterwards  replaced  by  the 
alphabetic  numerals,  in  which  the  letters  of  the  Greek  alphabet  were 
used,  together  with  three  strange  and  antique  letters  ^,  9 ,  and  T^ , 
and  the  symbol  M.  This  change  was  decidedly  for  the  worse,  for  the 
old  Attic  numerals  were  less  burdensome  on  the  memory,  inasmuch 

1  H.  Hankel,  Die  Entwickclung  der  Mathetnafik  in  den  Ictzten  Jahrhundcrtcn, 
Tubingen,  1884,  p.  16, 


GREEK  ARITHMETIC  AND  ALGEBRA  53 

as  they  contained  fewer  symbols  and  were  better  adapted  to  show 
forth  analogies  in  numerical  operations.  The  following  table  shov/s 
the  Greek  alphabetic  numerals  and  their  respective  values: — 

ajS'ySestv^'-        ''       ^       /*       "       ^        ^       ^       ' 

I     2     3     4     5     6     7     8     9     10     20     30     40     50     60     70     So     90 

par  V  ^  X  "P  "^  V^  '^  '^  'V  ^^^• 
100    200    300    400    500    600    700    800    900     1000    2000    3000 

/3  7 

M        M         M    etc. 

10,000  20,000  30,000 

It  will  be  noticed  that  at  1000,  the  alphabet  is  begun  over  again, 
but,  to  prevent  confusion,  a  stroke  is  now  placed  before  the  letter 
and  generally  somewhat  below  it.  A  horizontal  line  drawn  over  a 
number  served  to  distinguish  it  more  readily  from  words.  The  co- 
efficient for  M  was  sometimes  placed  before  or  behind  instead  of  over 
the  M.  Thus  43,678  was  written  SM.yxoT/.  It  is  to  be  observed  that 
the  Greeks  had  no  zero. 

Fractions  were  denoted  by  first  writing  the  numerator  marked  with 
an  accent,  then  the  denominator  marked  with  two  accents  and  written 
twice.  Thus,  ty'K^"K^"  =  i|.  In  case  of  fractions  having  unity  for 
the  numerator,  the  a'  was  omitted  and  the  denominator  was  written 
only  once.    Thus  /a8"  =  ^-^. 

The  Greeks  had  the  name  epimorion  for  the  ratio  ^^-j^.  Archytas 
proved  the  theorem  that  if  an  epimorion  ^  is  reduced  to  its  lowest 
terms  -,  then  v  =  /*+!.    This  theorem  is  found  later  in  the  musical 

V 

writings  of  Euclid  and  of  the  Roman  Boethius.  The  Euclidean  form 
of  arithmetic,  without  perhaps  the  representation  of  numbers  by  lines, 
existed  as  early  as  the  tune  of  Archytas.  1 

Greek  writers  seldom  refer  to  calculation  with  alphabetic  numerals. 
Addition,  subtraction,  and  even  multiplication  were  prolxably  per- 
formed on  the  abacus.  Expert  mathematicians  may  have  used  the 
symbols.  Thus  Eutocius,  a  commentator  of  the  sixth  century  after 
Christ,  gives  a  great  many  multiplications  of  which  the  following  is 


1  P.  Tannery  in  Bihlinthna  matliemalica,  3  S.,  Vol.  \T!,  iqo^.  p.  228 
''  }.  Govv,  (ii>.  <■//.,  p.  50. 


54  A  HlSrORY  OF  MATHEMATICS 

— V—  The  operation  is  explained  suf- 

°"^^  265  ficiently  by  the  modern  numerals 

o-ie  265  appended.       In    case    of    mixed 

Y^a  numbers,    the    process    was    still 

MM;/8,a     40000,    12000,    1000        more  clumsy.    Divisions  are  found 

a      in   Theon    of    Alexandria's   com- 

M,^^y-j^T     12000,      3600,      300        mentary    on    the    Almagest.     As 
,aTKe       ic>o^j        3^°;        25         might  be  expected,  the  process  is 

-y long  and  tedious. 

M  <tkT     "0^2-  ^^  have  seen  in  geometry  that 

the  more  advanced  mathematicians 
frequently  had  occasion  to  extract  the  square  root.  Thus  Archimedes 
in  his  Mensuration  of  the  Circle  gives  a  lar^^e  number  of  square  roots. 
He  states,  for  instance,  that  -\/^<-VfV-  ^^^*^  'v/o^xll'  ^^^the  gives  no 
clue  to  the  method  by  Avhich  he  obtained  these  approximations.  It 
is  not  improbable  that  the  earlier  Greek  mathematicians  found  the 
square  root  by  trial  only.  Eutocius  says  that  the  method  of  extracting 
it  was  given  by  Heron,  Pappus,  Theon,  and  other  commentators  on 
the  Almagest.  Theon's  is  the  only  one  of  these  methods  known  to  us. 
It  is  the  same  as  the  one  used  nowadays,  except  that  sexagesimal 
fractions  are  employed  in  place  of  our  decimals.  What  the  mode  of 
procedure  actually  was  when  sexagesimal  fractions  Avere  not  used,  has 
been  the  subject  of  conjecture  on  the  part  of  numerous  modern  writers. 
Of  interest,  in  connection  with  arithmetical  symbolism,  is  the  Sand- 
Counter  (Arenarius),  an  essay  addressed  by  Archimedes  to  Gelon, 
king  of  Syracuse.  In  it  Archimedes  shows  that  people  are  in  error  who 
think  the  sand  cannot  be  counted,  or  that  if  it  can  be  counted,  the 
number  cannot  be  expressed  by  arithmetical  symbols.  He  shows  that 
the  number  of  grains  in  a  heap  of  sand  not  only  as  large  as  the  whole 
earth,  but  as  large  as  the  entire  universe,  can  be  arithmetically  ex- 
pressed. Assuming  that  10,000  grains  of  sand  suffice  to  make  a  little 
solid  of  the  magnitude  of  a  po]:>py-seed,  and  that  the  diameter  of  a 
poppy-seed  be  not  smaller  than  -^  part  of  a  finger's  breadth;  assuming 
further,  that  the  diameter  of  the  universe  (supposed  to  extend  to  the 
sun)  be  less  than  10,000  diameters  of  the  earth,  and  that  the  latter 
be  less  than  i  ,000,000  stadia,  Archimedes  finds  a  number  which  w^ould 
exceed  the  number  of  grains  of  sand  in  the  sphere  of  the  universe. 
He  goes  on  even  further.  Supposing  the  universe  to  reach  out  to  the 
fixed  stars,  he  finds  that  the  sphere,  having  the  distance  from  the 
earth's  centre  to  the  fixed  stars  for  its  radius,  would  contain  a  number 
of  grains  of  sand  less  than  1000  myriads  of  the  eighth  octad.  In  our 
notation,  this  number  would  be  lo^s  or  i  with  63  ciphers  after  it.  It 
can  hardly  be  doubted  that  one  object  which  Archimedes  had  in  view 
in  making  this  calculation  was  the  improvement  of  the  Greek  sym- 
bolism. It  is  not  known  whether  he  invented  some  short  notation  by 
which  to  represent  the  above  number  or  not. 


GREEK  ARITHMETIC  AND  ALGEBRA 


55 


We  judge  from  fragments  in  the  second  book  of  Pai){)us  that  Apol- 
lonius  proposed  an  improvement  in  the  Greek  method  of  writing 
numbers,  but  its  nature  we  do  not  know.  Thus  we  see  that  the  Greeks 
never  possessed  the  boon  of  a  clear,  comprehensive  symbolism.  The 
honor  of  gi\-ing  such  to  the  world  was  reserved  by  the  iron}-  of  fate 
lor  a  nameless  Indian  of  an  unknown  time,  and  we  know  not  whom  to 
thank  for  an  invention  of  such  importance  to  the  general  progress  of 
intelligence. 1 

Passing  from  the  subject  of  logistic  a  to  that  of  arithmctica,  our  at- 
tention is  first  drawn  to  the  science  of  numbers  of  Pythagoras.  Before 
founding  his  school,  Pythagoras  studied  for  many  years  under  the 
Egyptian  priests  and  familiarised  himself  with  Egyptian  mathematics 
and  mysticism.  If  he  ever  was  in  Babylon,  as  some  authorities  claim, 
he  may  have  learned  the  sexagesimal  notation  in  use  there;  he  may 
have  picked  up  considerable  knowledge  on  the  theory  of  proportion, 
and  may  have  found  a  large  number  of  interesting  astronomical 
observations.  Saturated  with  that  speculative  spirit  then  pervading 
the  Greek  mind,  he  endeavored  to  discover  some  principle  of  homo- 
geneity in  the  universe.  Before  him,  the  philosophers  of  the  Ionic 
school  had  sought  it  in  the  matter  of  things;  Pythagoras  looked  for 
it  in  the  structure  of  things.  He  observed  various  numerical  relations 
oi  analogies  between  numbers  and  the  phenomena  of  the  universe. 
Being  convinced  that  it  was  in  numbers  and  their  relations  that  he 
was  to  find  the  foundation  to  true  philosophy,  he  proceeded  to  trace 
the  origin  of  all  things  to  numbers.  Thus  he  observed  that  musical 
strings  of  equal  length  stretched  by  weights  having  the  proportion  of 
i- 1, 1-,  produced  intervals  which  were  an  octave,  a  fifth,  and  a  fourth. 
Harmony,  therefore,  depends  on  musical  proportion;  it  is  nothing  but 
a  mysterious  numerical  relation.  Where  harmony  is,  there  are 
numbers.  Hence  the  order  and  beauty  of  the  universe  have  their 
origin  in  numbers.  There  are  seven  intervals  in  the  musical  scale, 
and  also  seven  planets  crossing  the  heavens.  The  same  numerical 
relations  which  underlie  the  former  must  underlie  the  latter.  But 
where  numbers  are,  there  is  harmony.  Hence  his  spiritual  ear  dis- 
cerned in  the  planetary  motions  a  wonderful  "harmony  of  the  spheres." 
The  Pythagoreans  invested  particular  numbers  with  extraordinary 
attributes.  Thus  one  is  the  essence  of  things;  it  is  an  absolute  number; 
hence  the  origin  of  all  numbers  and  so  of  all  things.  Four  is  the  most 
perfect  number,  and  was  in  some  mystic  way  conceived  to  correspond 
to  the  human  soul.  Philolaus  believed  that  5  is  the  cause  of  color,  6  of 
cold,  7  of  mind  and  health  and  fight,  8  of  love  and  friendship. 2  In 
Plato's  works  are  evidences  of  a  similar  belief  in  religious  relations  of 
numbers.    Even  Aristotle  referred  the  virtues  to  numbers. 

Enough  has  been  said  about  these  mystic  speculations  to  show 
what  lively  interest  in  mathematics  they  must  have  created  and 

1  J.  Gow,  op.  cit.,  p.  63c  -J.  Gow,  op.  cit.,  p.  69. 


56  A  HISTORY  OF  MATHEMATICS 

maintained.  Avenues  of  mathematical  inquiry  were  opened  up  by 
them  which  otherwise  would  probably  have  remained  closed  at  that 
time. 

The  Pythagoreans  classified  numbers  into  odd  and  even.  They 
observed  that  the  sum  of  the  series  of  odd  numbers  from  i  to  2n+i 
was  always  a  complete  square,  and  that  by  addition  of  the  even  num- 
bers arises  the  series  2,  6,  12,  20,  in  which  every  number  can  be  de- 
composed into  two  factors  differing  from  each  other  by  unity.  Thus, 
6=2.3,  12  =  3.4,  etc.  These  latter  numbers  were  considered  of 
sufiScient  importance  to  receive  the  separate  name  of  heteromecic  (not 

equilateral).     Numbers  of  the  form  were  called  triangular, 

because  they  could  always  be  arranged  thus,  .•,*,•♦  Nimibers  which 
were  equal  to  the  sum  of  all  their  possible  factors,  such  as  6,  28,  496, 
were  called  perfect;  those  exceeding  that  sum,  excessive;  and  those 
which  were  less,  defective.  Amicable  numbers  were  those  of  which 
each  w^as  the  sum  of  the  factors  in  the  other.  Aluch  attention  was 
paid  by  the  Pythagoreans  to  the  subject  of  proportion.  The  quan- 
tities a,  b,  c,  d  were  said  to  be  in  arithmetical  proportion  when  a  —  b  = 
c  —  d;  in  geometrical  proportion,  when  a:b  =  c:d;  in  harmonic  propor- 
tion, when  a—b:b  —  c=a:c.     It  is  probable  that  the  Pythagoreans 

were   also   familiar   with   the   musical   proportion   a: ~~Ta'^' 

lamblichus  says  that  Pythagoras  introduced  it  from  Babylon. 

In  connection  with  arithmetic,  Pythagoras  made  extensive  investi- 
gations into  geometry.  He  believed  that  an  arithmetical  fact  had 
its  analogue  in  geometry,  and  vice  versa.  In  connection  with  his 
theorem  on  the  right  triangle  he  devised  a  rule  by  which  integral 
numbers  could  be  found,  such  that  the  sum  of  the  squares  of  two  of 
them  equalled  the  square  of  the  third.    Thus,  take  for  one  side  an  odd 

number  (2^4-1);   then =  2n^-\-2n=t}ie  other  side,  and 

(2»2-{-  2n+ 1)  =  hypotenuse.  If  2n-\- 1  =  9,  then  the  other  two  numbers 
are  40  and  41.  But  this  rule  only  applies  to  cases  in  which  the  hy- 
potenuse differs  from  one  of  the  sides  by  i.  In  the  study  of  the  right 
triangle  there  doubtless  arose  questions  of  puzzUng  subtlety.  Thus, 
given  a  number  equal  to  the  side  of  an  isosceles  right  triangle,  to  find 
the  number  which  the  hypotenuse  is  equal  to.  The  side  may  have 
been  taken  equal  to  i,  2,  |,  |,  or  any  other  number,  yet  in  every  in- 
stance all  efforts  to  find  a  number  exactly  equal  to  the  hypotenuse 
must  have  remained  fruitless.  The  problem  may  have  been  attacked 
again  and  again,  until  finally  "  some  rare  genius,  to  whom  it  is  granted, 
during  some  happy  moments,  to  soar  with  eagle's  flight  above  the 
level  of  human  thinking,"  grasped  the  happy  thought  that  this  prob- 


GREEK  ARITHMETIC  AND  ALGEBR.A 


57 


lem  cannot  be  solved.  In  some  such  manner  probably  arose  the  theory 
of  irrational  quantities,  which  is  attributed  by  Eudemus  to  the  Pytha- 
goreans. It  was  indeed  a  thought  of  extraordinary  boldness,  to  as- 
sume that  straight  lines  could  exist,  differing  from  one  another  not 
only  in  length, — that  is,  in  quantity, — but  also  in  a  quality,  which, 
though  real,  was  absolutely  invisible. ^  Need  we  wonder  that  the 
Pythagoreans  saw  in  irrationals  a  deep  mystery,  a  symbol  of  the  un- 
speakable? We  are  told  that  the  one  who  first  divulged  the  theory  of 
irrationals,  which  the  Pythagoreans  kept  secret,  perished  in  conse- 
quence in  a  shipwreck,  "for  the  unspeakable  and  invisible  should 
always  be  kept  secret."  Its  discovery  is  ascribed  to  Pythagoras,  but 
we  must  remember  that  all  important  Pythagorean  discoveries  were, 
according  to  Pythagorean  custom,  referred  back  to  him.  The  first 
incommensurable  ratio  known  seems  to  have  been  that  of  the  side 
of  a  square  to  its  diagonal,  as  i  :\/2.  Theodonis  of  Cyrene  added  to 
this^the  fact  that  th_e_ sides  of  squares  represented  in  length  by  y/ t„ 
V5,  etc.,  up  to  V  17,  and  Theaetetus,  that  the  sides  of  any  square, 
represented  by  a  surd,  are  incommensurable  with  the  linear  unit. 
Euclid  (about  300  b.  c),  in  his  Elements,  X,  9,  generalised  still  further: 
Two  magnitudes  whose  squares  are  (or  are  not)  to  one  another  as  a 
square  number  to  a  square  number  are  commensurable  (or  incom- 
mensurable), and  conversely.  In  the  tenth  book,  he  treats  of  incom- 
mensurable quantities  at  length.  He  investigates  every  possible 
variety  of  lines  which  can  be  represented  by  -\/\/o='=  Vb,  a  and  b 
representing  two  commensurable  lines,  and  obtains  25  species.  Every 
individual  of  every  species  is  incommensurable  with  all  the  individuals 
of  every  other  species.  "This  book,"  says  De  Morgan,  "has  a  com- 
pleteness which  none  of  the  others  (not  even  the  fifth)  can  boast  of; 
and  we  could  almost  suspect  that  Euclid,  having  arranged  his  ma- 
terials in  his  own  mind,  and  having  completely  elaborated  the  tenth 
book,  wrote  the  preceding  books  after  it,  and  did  not  live  to  revise 
them  thoroughly."  2  The  theory  of  incommensurables  remained 
where  Euclid  left  it,  till  the  fifteenth  century. 

If  it  be  recalled  that  the  early  Egyptians  had  some  familiarity  with 
quadratic  equations,  it  is  not  surprising  if  similar  knowledge  is  dis- 
played by  Greek  writers  in  the  time  of  Pythagoras.  Hippocrates,  in 
the  fifth  century  b.  c,  when  working  on  the  areas  of  lunes,  assumes 
the  geometrical  equivalent  of  the  solution  of  the  quadratic  equation 
•v-+Vf  ax=a-.  The  complete  geometrical  solution  was  given  by 
Euclid  in  his  Elements,  VI,  27-29.  He  solves  certain  types  of  quad- 
ratic equations  geometrically  in  Book  II,  5,  6,  11. 

'  H.  Hankel,  Zur  Geschichle  der  Malhematik  in  MiUelaller  utid  AUerthum,  1874, 
p.  102. 

-A.  De  Morgan,  "Eucleides"  in  Smith's  Diclionarv  of  Greek  and  Roman  Biog. 
and  Mylji 


58  A  HISTORY  OF  MATHEMATICS 

Euclid  devotes  the  seventh,  eighth,  and  ninth  books  of  his  Elements 
to  arithmetic.  Exactly  how  much  contained  in  these  books  is  Euclid's 
own  invention,  and  how  much  is  borrowed  from  his  predecessors,  we 
have  no  means  of  knowing.  Without  doubt,  much  is  original  with 
Euclid.  The  seventh  book  begins  with  twenty-one  definitions.  All 
except  that  for  "prime"  numbers  are  known  to  have  been  given  by 
the  Pythagoreans.  Next  follows  a  process  for  finding  the  G.  C.  D. 
of  two  or  more  numbers.  The  eighth  book  deals  with  numbers  in  con- 
tinued proportion,  and  with  the  mutual  relations  of  squares,  cubes, 
and  plane  numbers.  Thus,  XXII,  if  three  numbers  are  in  continued 
proportion,  and  the  first  is  a  square,  so  is  the  third.  In  the  ninth  book, 
the  same  subject  is  continued.  It  contains  the  proposition  that  the 
number  of  primes  is  greater  than  any  given  number. 

After  the  death  of  Euclid,  the  theory  of  numbers  remained  almost 
stationary  for  400  years.  Geometry  monopolised  the  attention  of  all 
Greek  mathematicians.  Only  two  are  known  to  have  done  work  in 
arithmetic  worthy  of  mention.  Eratosthenes  (275-194  b.  c.)  invented 
a  "sieve"  for  finding  prime  numbers.  All  composite  numbers  are 
"sifted"  out  in  the  following  manner:  Write  down  the  odd  numbers 
from  3  up,  in  succession.  By  striking  out  every  third  number  after 
the  3,  we  remove  all  multiples  of  3.  By  striking  out  every  fifth  num- 
ber after  the  5,  we  remove  all  multiples  of  5.  In  this  way,  by  rejecting 
multiples  of  7,  11,  13,  etc.,  we  have  left  prime  numbers  only.  Hyp- 
sicles  (between  200  and  100  b.  c.)  worked  at  the  subjects  of  polygonal 
numbers  and  arithmetical  progressions,  which  Euclid  entirely  neg- 
lected. In  his  work  on  "risings  of  the  stars,"  he  showed  (i)  that  in 
an  arithmetical  series  of  2n  terms,  the  sum  of  the  last  n  terms  exceeds 
the  sum  of  the  first  n  by  a  multiple  of  «2;  (2)  that  in  such  a  series  of 
2n-\- 1  terms,  the  sum  of  the  series  is  the  number  of  terms  multiplied 
by  the  middle  term;  (3)  that  in  such  a  series  of  2W  terms,  the  sum  is 
half  the  number  of  terms  multiplied  by  the  two  middle  terms.^ 

For  two  centuries  after  the  time  of  Hypsicles,  arithmetic  disappears 
from  history.  It  is  brought  to  light  again  about  100  a.  d.  by  Ni- 
comachus,  a  Neo-Pythagorean,  who  inaugurated  the  final  era  of  Greek 
mathematics.  From  now  on,  arithmetic  was  a  favorite  study,  while 
geometry  was  neglected.  Nicomachus  wrote  a  work  entitled  In- 
troductio  Arithmetica,  which  was  very  famous  in  its  day.  The  great 
number  of  commentators  it  has  received  vouch  for  its  popularity. 
Boethius  translated  it  into  Latin.  Lucian  could  pay  no  higher  com- 
pliment to  a  calculator  than  this:  "You  reckon  like  Nicomachus  of 
Gerasa."  The  Introdnctio  Arithmetica  was  the  first  exhaustive  work 
in  which  arithmetic  was  treated  quite  independently  of  geometry. 
Instead  of  drawing  lines,  like  Euclid,  he  illustrates  things  by  real 
numbers.  To  be  sure,  in  his  book  the  old  geometrical  nomenclature  is 
retained,  but  the  method  is  inductive  instead  of  deductive.  "Its  sole 
^  J.  Gow,  op.  cii.,  p.  87. 


GREEK  ARITHMETIC  AND  ALGEBRA 


i9 


business  is  classification,  and  all  its  classes  are  dcrixed  from,  and 
exhibited  by,  actual  numbers."  The  work  contains  few  results  that 
are  really  original.  We  mention  one  important  pro[)osition  which  is 
probably  the  author's  own.  He  states  that  cubical  numbers  are  al- 
ways equal  to  the  sum  of  successive  odd  numbers.  Thus,  8  =  2^  = 
3+5,  27  =  33=7-1-9+11,  64=43=13+15-1-17+19,  and  so  on.  This 
theorem  was  used  later  for  finding  the  sum  of  the  cubical  numbers 
themselves.  Theon.  of  Smyrna  is  the  author  of  a  treatise  on  "the 
mathematical  rules  necessary  for  the  study  of  Plato."  The  work  is 
ill  arranged  and  of  little  merit.  Of  interest  is  the  theorem,  that  every 
square  number,  or  that  number  minus  i,  is  divisible  by  3  or  4  or  both. 
A  remarkable  discovery  is  a  proposition  given  by  lamblichus  in  his 
treatise  on  Pythagorean  philosophy.  It  is  founded  on  the  observation 
that  the  Pythagoreans  called  i,  10,  100,  1000,  units  of  the  first,  second, 
third,  fourth  "course"  respectively.  The  theorem  is  this:  If  we  add 
any  three  consecutive  numbers,  of  which  the  highest  is  divisible  by  3, 
then  add  the  digits  of  that  sum,  then,  again,  the  digits  of  that  sum, 
and  so  on,  the  final  sum  will  be  6.  Thus,  61+62+63=  186, 1+8+6  = 
15)  1+5  =  6.  This  discovery  was  the  more  remarkable,  because  the 
ordinary^  Greek  numerical  s^Tnbolism  was  much  less  likely  to  suggest 
any  such  property  of  numbers  than  our  "Arabic"  notation  would 
have  been. 

Hippolytus,  who  appears  to  have  been  bishop  at  Tortus  Romae  in 
Italy  in  the  early  part  of  the  third  century,  must  be  mentioned  for  the 
giving  of  "proofs"  by  casting  out  the  9's  and  the  7's. 

The  works  of  Nicomachus,  Theon  of  Smyrna,  Thymaridas,  and 
others  contain  at  times  investigations  of  subjects  which  are  really 
algebraic  in  their  nature.  Thymaridas  in  one  place  uses  the  Greek, 
word  meaning  "unknown  quantity"  in  a  way  which  would  lead  one 
to  believe  that  algebra  was  not  far  distant.  Of  interest  in  tracing  the 
invention  of  algebra  are  the  arithmetical  epigrams  in  the  Palatine 
Anthology,  which  contain  about  fifty  problems  leading  to  linear  equa- 
tions. Before  the  introduction  of  algebra  these  problems  were  pro- 
pounded as  puzzles.  A  riddle  attributed  to  Euclid  and  contained  in 
the  Anthology  is  to  this  effect:  A  mule  and  a  donkey  were  walking 
along,  laden  with  corn.  The  mule  says  to  the  donkey,  "If  you  gave 
me  one  measure,  I  should  carry  twice  as  much  as  you.  If  I  gave  you 
one,  we  should  both  carry  equal  burdens.  Tell  me  their  burdens,  O 
most  learned  master  of  geometry."  ^ 

It  will  be  allowed,  says  Gow,  that  this  problem,  if  authentic,  was 
not  beyond  Euclid,  and  the  appeal  to  geometry  smacks  of  antiquity. 
A  far  more  difficult  puzzle  was  the  famous  "cattle-problem,"  which 
Archimedes  propounded  to  the  Alexandrian  mathematicians.  The 
problem  is  indeterminate,  for  from  only  seven  equations,  eight  un- 
known quantities  in  integral  numbers  are  to  be  found.  It  may  be 
^  J.  Gow,  op.  cit.,  p.  99. 


6o  A  HISTORY  OF  MATHEMATICS 

stated  thus:  The  sun  had  a  herd  of  bulls  and  cows,  of  different  colors, 
(i)  Of  Bulls,  the  white  (IF)  were,  in  number,  (|+|)  of  the  blue  (B) 
and  yellow  (F):  the  B  were  (^+|)  of  the  Y  and  piebald  (P):  the  P 
were  (■g-+4)  of  the  TF  and  Y.  (2)  Of  Cows,  which  had  the  same  colors 
(w,  b,  y,  p), 

^=(H4)  iB+b):b=a+l)  (P+p):p=a+^)  (Y+y):y  =  a-h^). 

(W+w). 

Find  the  number  of  bulls  and  cows.^  This  leads  to  high  numbeis, 
but,  to  add  to  its  complexity,  the  conditions  are  superadded  that 
W+B  =  a  square,  and  PH-Y=a  triangular  nimiber,  leading  to  an  in- 
determinate equation  of  the  second  degree.  Another  problem  in  the 
Anthology  is  quite  familiar  to  school-boys:  "Of  four  pipes,  one  fills  the 
cistern  in  one  day,  the  next  in  two  days,  the  third  in  three  days,  the 
fourth  in  four  days:  if  all  run  together,  how  soon  will  they  fill  the 
cistern?"  A  great  many  of  these  problems,  puzzling  to  an  arith- 
metician, would  have  been  solved  easily  by  an  algebraist.  They  be- 
came very  popular  about  the  time  of  Diophantus,  and  doubtless  acted 
as  a  powerful  stimulus  on  his  mind. 

Diophantus  was  one  of  the  last  and  most  fertile  mathematicians  of 
the  second  Alexandrian  school.  He  flourished  about  250  a.  d.  His 
age  was  eighty-four,  as  is  known  from  an  epitaph  to  this  effect:  Dio- 
phantus passed  ^  of  his  life  in  childhood.  -^  in  youth,  and  1  more  as 
a  bachelor;  five  years  after  his  marriage  was  born  a  son  who  died  four 
years  before  his  father,  at  half  his  father's  age.  The  place  of  nativity 
and  parentage  of  Diophantus  are  unknown.  If  his  works  were  not 
written  in  Greek,  no  one  would  think  for  a  moment  that  they  were 
the  product  of  Greek  mind.  There  is  nothing  in  his  works  that 
reminds  us  of  the  classic  period  of  Greek  mathematics.  His  were  al- 
most entirely  new  ideas  on  a  new  subject.  In  the  circle  of  Greek 
mathematicians  he  stands  alone  in  his  specialty.  Except  for  him, 
we  should  be  constrained  to  say  that  among  the  Greeks  algebra  was 
almost  an  unknown  science. 

Of  his  works  we  have  lost  the  Porisms,  but  possess  a  fragment  of 
Polygonal  Numbers,  and  seven  books  of  his  great  work  on  Arithmetica, 
said  to  have  been  written  in  13  books.  Recent  editions  of  the  Arith- 
metica were  brought  out  by  the  indefatigable  historians,  P.  Tannery 
and  T.  L.  Heath,  and  by  G.  Wertheim. 

If  we  except  the  Ahmes  papyrus,  which  contains  the  first  sugges- 
tions of  algebraic  notation,  and  of  the  solution  of  equations,  then  his 
Arithmetica  is  the  earliest  treatise  on  algebra  now  extant.  In  this  work 
is  introduced  the  idea  of  an  algebraic  equation  expressed  in  algebraic 
symbols.  His  treatment  is  purely  analytical  and  completely  divorced 
from  geometrical  methods.  He  states  that  "a  number  to  be  sub- 
^  J.  Gow,  op.  cit.,  p.  99. 


GREEK  ARITHMETIC  AND  ALGEBRA  6i 

tracted,  multiplied  by  a  number  to  be  subtracted,  gives  a  number  to 
be  added."  This  is  applied  to  the  multiplication  of  differences,  such 
as  (v— i)  (.V— 2).  It  must  be  remarked,  that  Diophantus  had  no 
notion  whatever  of  negative  numbers  standing  by  themselves.  All 
he  knew  were  differences,  such  as  (2.r—  10),  in  which  2x  could  not  be 
smaller  than  10  without  leading  to  an  absurdity.  He  appears  to  be 
the  first  who  could  perform  such  operations  as  {x-  i)x(x-  2)  without 
reference  to  geometry.  Such  identities  as  {a  -[-b)-  =a^+2ab  +b^,  which 
with  Euclid  appear  in  the  elevated  rank  of  geometric  theorems,  are 
with  Diophantus  the  simplest  consequences  of  the  algebraic  laws  of 
operation.  His  sign  for  subtraction  was  ^,  for  equality  t.  For  un- 
known quantities  he  had  only  one  symbol,  s.  He  had  no  sign  for 
addition  except  juxtaposition.  Diophantus  used  but  few  symbols, 
and  sometimes  ignored  even  these  by  describing  an  operation  in  words 
when  the  symbol  would  have  answered  just  as  well. 

In  the  solution  of  simultaneous  equations  Diophantus  adroitly 
managed  with  only  one  symbol  for  the  unknown  quantities  and  ar- 
rived at  answers,  most  commonly,  by  the  method  of  tentative  assump- 
tion, which  consists  in  assigning  to  some  of  the  unknown  quantities 
preliminary  values,  that  satisfy  only  one  or  two  of  the  conditions. 
These  values  lead  to  expressions  palpably  wrong,  but  which  generally 
suggest  some  stratagem  by  which  values  can  be  secured  satisfying 
all  the  conditions  of  the  problem. 

Diophantus  also  solved  determinate  equations  of  the  second  degree. 
Such  equations  were  solved  geometrically  by  Euclid  and  Hippocrates. 
Algebraic  solutions  appear  to  have  been  found  by  Heron  of  Alexandria, 
who  gives  8|  as  an  approximate  answer  to  the  equation  i44.t(i4  — x)  = 
6720.  In  the  Geometry,  doubtfully  attributed  to  Heron,  the  solution  of 
the  equation  ^x^-{-?-^x=2i2  is  practically  stated  in  the  form  x  = 

■ ^^ ^^-^ -.    Diophantus  nowhere  goes  through  with  the 

whole  process  of  solving  quadratic  equations;  he  merely  states  the 
result.  Thus,  "84.^24-7.-^=7,  whence  x  is  found  =|-."  From  partial 
explanations  found  here  and  there  it  appears  that  the  quadratic  equa- 
tion was  so  written  that  all  terms  were  positive.  Hence,  from  the  point 
of  view  of  Diophantus,  there  were  three  cases  of  equations  with  a 
positive  root:  ax'+bx  =c,  ax^  =bx+c,  ax'^+c  =bx,  each  case  requiring 
a  rule  slightly  different  from  the  other  two.  Notice  he  gives  only  one 
root.  His  failure  to  observe  that  a  quadratic  equation  has  two  roots, 
even  when  both  roots  are  positive,  rather  surprises  us.  It  must  be 
remembered,  however,  that  this  same  inability  to  perceive  more  than 
one  out  of  the  several  solutions  to  which  a  problem  may  point  is  com- 
mon to  all  Greek  mathematicians.  Another  point  to  be  observed 
is  that  he  never  accepts  as  an  answer  a  quantity  which  is  negative 
or  irrational. 


62  A  HISTORY  OF  MATHEMATICS 

Diophantus  devotes  only  the  first  book  of  his  Arithmetica  to  the 
solution  of  determinate  equations.  The  remaining  books  extant 
treat  mainly  of  indeterminate  quadratic  equations  of  the  form  Ax^  + 
Bx+C  =y-,  or  of  two  simultaneous  equations  of  the  same  form.  He 
considers  several  but  not  all  the  possible  cases  which  may  arise  in 
these  equations.  The  opinion  of  Nesselmann  on  the  method  of  Dio- 
phantus, as  stated  by  Gow,  is  as  follows:  "  (i)  Indeterminate  equations 
of  the  second  degree  are  treated  completely  only  when  the  quadratic 
or  the  absolute  term  is  wanting:  his  solution  of  the  equations  Ax^  + 
C  =y^  and  Ax^+Bx+C  =y^  is  in  many  respects  cramped.  (2)  For 
the  'double  equation'  of  the  second  degree  he  has  a  definite  rule  only 
when  the  quadratic  term  is  wanting  in  both  expressions:  even  then 
his  solution  is  not  general.  More  complicated  expressions  occur  only 
under  specially  favourable  circumstances."  Thus,  he  solves  Bx+C^ 
=3/2^  Bix+Ci-=yi-. 

The  extraordinary  abiUty  of  Diophantus  lies  rather  in  another  di- 
rection, namely,  in  his  wonderful  ingenuity  to  reduce  all  sorts  of 
equations  to  particular  forms  which  he  knows  how  to  solve.  Very 
great  is  the  variety  of  problems  considered.  The  130  problems  found 
in  the  great  work  of  Diophantus  contain  over  50  different  classes  of 
problems,  which  are  strung  together  without  any  attempt  at  classi- 
fication. But  still  more  multifarious  than  the  problems  are  the  solu- 
tions. General  methods  are  almost  unknown  to  Dipohantus.  Each 
problem  has  its  own  distinct  method,  which  is  often  useless  for  the 
most  closely  related  problems.  "  It  is,  therefore,  difficult  for  a  modern, 
after  studying  100  Diophantine  solutions,  to  solve  the  loist."  This 
statement,  due  to  Hankel,  is  somewhat  overdrawn,  as  is  shown  by 
Heath.i 

That  which  robs  his  work  of  much  of  its  scientific  value  is  the 
fact  that  he  always  feels  satisfied  with  one  solution,  though  his  equa- 
tion may  admit  of  an  indefinite  number  of  values.  Another  great 
defect  is  the  absence  of  general  methods.  Modern  mathematicians, 
such  as  L.  Euler,  J.  Lagrange,  K.  F.  Gauss,  had  to  begin  the  study  of 
indeterminate  analysis  anew  and  received  no  direct  aid  from  Dio- 
phantus in  the  formulation  of  methods.  In  spite  of  these  defects 
we  cannot  fail  to  admire  the  work  for  the  wonderful  ingenuity  ex- 
hibited therein  in  the  solution  of  particular  equations. 

^T.  L.  Heath,  Diophantus  of  Alexandria,  2  Ed.,  Cambridge,  1910,  pp.  54-97. 


THE  ROMANS 

Nowhere  is  the  contrast  between  the  Greek  and  Roman  minds 
shown  forth  more  distinctly  than  in  their  attitude  toward  the  mathe- 
matical science.  The  sway  of  the  Greek  was  a  flowering  time  for 
mathematics,  but  that  of  the  Roman  a  period  of  sterility.  In  philos- 
ophy, poetry,  and  art  the  Roman  was  an  imitator.  But  in  mathe- 
matics he  did  not  even  rise  to  the  desire  for  imitation.  The  mathe- 
matical fruits  of  Greek  genius  lay  before  him  untasted.  In  him  a 
science  which  had  no  direct  bearing  on  practical  life  could  awake  no 
interest.  As  a  consequence,  not  only  the  higher  geometry  of  Archi- 
medes and  Apollonius,  but  even  the  Elements  of  Euclid,  were  neglected. 
What  little  mathematics  the  Romans  possessed  did  not  come  altogether 
from  the  Greeks,  but  came  in  part  from  more  ancient  sources.  Exactly 
where  and  how  some  of  it  originated  is  a  matter  of  doubt.  It  seems 
most  probable  that  the  "Roman  notation,"  as  well  as  the  early 
practical  geometry  of  the  Romans,  came  from  the  old  Etruscans, 
who,  at  the  earliest  period  to  which  our  knowledge  of  them  extends, 
inhabited  the  district  between  the  Arno  and  Tiber. 

Li\y  tells  us  that  the  Etruscans  were  in  the  habit  of  representing 
the  number  of  years  elapsed,  by  driving  yearly  a  nail  into  the  sanc- 
tuary of  Minerva,  and  that  the  Romans  continued  this  practice.  A 
less  primitive  mode  of  designating  numbers,  presumably  of  Etruscan 
origin,  was  a  notation  resembling  the  present  "Roman  notation." 
This  system  is  noteworthy  from  the  fact  that  a  principle  is  involved 
in  it  which  is  rarely  met  with  in  others,  namely,  the  principle  of  sub- 
traction. If  a  letter  be  placed  before  another  of  greater  value,  its 
value  is  not  to  be  added  to,  but  subtracted  from,  that  of  the  greater. 
In  the  designation  of  large  numbers  a  horizontal  bar  placed  over  a 
letter  was  made  to  increase  its  value  one  thousand  fold.  In  fractions 
the  Romans  used  the  duodecimal  system. 

Of  arithmetical  calculations,  the  Romans  employed  three  diflferent 
kinds:  Reckoning  on  the  fingers,  upon  the  abacus,  and  by  tables  pre- 
pared for  the  purpose.  1  Finger-symbolism  was  known  as  early  as  the 
time  of  King  Numa,  for  he  had  erected,  says  Pliny,  a  statue  of  the 
double-faced  Janus,  of  which  the  fingers  indicated  365  (355?),  the 
number  of  days  in  a  year.  Many  other  passages  from  Roman  authors 
point  out  the  use  of  the  fingers  as  aids  to  calculation.  In  fact,  a  finger- 
symbolism  of  practically  the  same  form  was  in  use  not  only  in  Rome, 
but  also  in  Greece  and  throughout  the  East,  certainly  as  early  as  the 
beginning  of  the  Christian  era,  and  continued  to  be  used  in  Europe 
1  M.  Cantor,  op  cit.,  Vol.  I,  3  Aufl.,  1907,  p.  526. 
63 


64 


A  HISTORY  OF  MATHEMATICS 


during  the  Middle  Ages.  We  possess  no  knowledge  as  to  where  or  when 
it  was  invented.  The  second  mode  of  calculation,  by  the  abacus,  was 
a  subject  of  elementary  instruction  in  Rome.  Passages  in  Roman 
writers  indicate  that  the  kind  of  abacus  most  commonly  used  was 
covered  with  dust  and  then  divided  into  columns  by  drawing  straight 
lines.  Each  column  was  supplied  with  pebbles  (calculi,  whence  "cal- 
culare"  and  "calculate")  which  served  for  calculation. 

The  Romans  used  also  another  kind  of  abacus,  consisting  of  a 
metallic  plate  having  grooves  with  movable  buttons.  By  its  use  all 
integers  between  i  and  9,999,999,  as  well  as  some  fractions,  could  be 
represented.    In  the  two  adjoining  figures  ^  the  lines  represent  grooves 


II  C  X   I  C  X   I 


ii 

,  < 

i  i 

1  ( 
)  ( 

I  ( 

i  i 

il 

^1 


li  c  X  r  c  X 


0  0  0  0 


and  the  circles  buttons.  The  Roman  numerals  indicate  the  value  of 
each  button  in  the  corresponding  groove  below,  the  button  in  the 
shorter  groove  above  having  a  fivefold  value.  Thus  If  =1,000,000; 
hence  each  button  in  the  long  left-hand  groove,  when  in  use,  stands 
for  1,000,000,  and  the  button  in  the  short  upper  groove  stands  for 
5,000,000.  The  same  holds  for  the  other  grooves  labelled  by  Roman 
numerals.  The  eighth  long  groove  from  the  left  (having  5  buttons) 
represents  duodecimal  fractions,  each  button  indicating  -^^,  while  the 
button  above  the  dot  means  ■^.  In  the  ninth  column  the  upper 
button  represents  ^,  the  middle  -^-g,  and  two  lower  each  ^.  Our 
first  figure  represents  the  positions  of  the  buttons  before  the  operation 
begins;  our  second  figure  stands  for  the  number  852  |  ~.  The  eye 
has  here  to  distinguish  the  buttons  in  use  and  those  left  idle.  Those 
counted  are  one  button  above  c  (=500),  and  three  buttons  below 
c  (  =  300) ;  one  button  above  x  (  =  50) ;  two  buttons  below  I  (  =  2) ;  four 
buttons  indicating  duodecimals  (  =  |);  and  the  button  for  ^■ 

Suppose  now  that  10,318  |  i  -^^  is  to  be  added  to  852  |  ^.  The 
operator  could  begin  with  the  highest  units,  or  the  lowest  units,  as  he 
pleased.    Naturally  the  hardest  part  is  the  addition  of  the  fractions. 

»  G.  Friedlein,  Die  Zahlzeichen  und  das  elementare  Rechnen  der  Griechen  utid  Rdmer, 
Erlangen,  1869,  Fig.  21.  Gottfried  Friedlein  (1828-1875)  was  "Rektor  der  Kgl. 
Studienanstalt  zu  Hof "  in  Bavaria. 


THE  ROMANS  65 


buttons  below  the  dot  were  used  to  indicate  the  sum  |  ~.  The  addi- 
tion of  8  would  bring  all  the  buttons  above  and  below  i  into  play, 
making  10  units.  Hence,  move  them  all  back  and  move  up  one  button 
in  the  groove  below  x.  Add  10  by  moving  up  another  of  the  buttons 
below  x;  add  300  to  800  by  moving  back  all  buttons  above  and  below 
c,  except  one  button  below,  and  moving  up  one  button  below  I;  add 
10,000  by  moving  up  one  button  below  x.  In  subtraction  the  operation 
was  similar. 

Multiplication  could  be  carried  out  in  several  ways.  In  case  of 
38  2  XT  times  25  i,  the  abacus  may  have  shown  successively  the  follow- 
ing values:  600  (=30.20),  760  (=600-1-20.8),  770  (=760-1-^.20), 
77o|^  (=770+2^.20),  92o|^  (=7701^+30.5),  960  {^  (=920  14 -f- 
8-5),  963  I  (=96ol4+^5),  963!  2T  (=963Hti-5),  973  ^^ 
(=963  i  ^+430),  976  A  A  (=973  I  ^+84),  976  i  ^  (=976 
T%  #T+i-i).  976  I  ^  T2  (  =976  i  ^+i.  A).^ 

In  division  the  abacus  was  used  to  represent  the  remainder  resulting 
from  the  subtraction  from  the  dividend  of  the  divisor  or  of  a  con- 
venient multiple  of  the  divisor.  The  process  was  complicated  and 
difficult.  These  methods  of  abacal  computation  show  clearly  how 
multiplication  or  division  can  be  carried  out  by  a  series  of  successive 
additions  or  subtractions.  In  this  connection  we  suspect  that  recourse 
was  had  to  mental  operations  and  to  the  multiplication  table.  Pos- 
sibly finger-multiplication  may  also  have  been  used.  But  the  multi- 
plication of  large  numbers  must,  by  either  method,  have  been  beyond 
the  power  of  the  ordinary  arithmetician.  To  obviate  this  difficulty, 
the  arithmetical  tables  mentioned  above  were  used,  from  which  the 
desired  products  could  be  copied  at  once.  Tables  of  this  kind  were 
prepared  by  Vidorius  of  Aquitania.  His  tables  contain  a  pecuHar 
notation  for  fractions,  which  continued  in  use  throughout  the  Middle 
Ages.  Victorius  is  best  known  for  his  canon  paschalis,  a.  rule  for  find- 
ing the  correct  date  for  Easter,  which  he  published  in  457  a.  d. 

Payments  of  interest  and  problems  in  interest  were  very  old  among 
the  Romans.  The  Roman  laws  of  inheritance  gave  rise  to  numerous 
arithmetical  examples.  Especially  unique  is  the  following:  A  dying 
man  wills  that,  if  his  wife,  being  with  child,  gives  birth  to  a  son,  the 
son  shall  receive  ^  and  she  i  of  his  estates;  but  if  a  daughter  is  born, 
she  shall  receive  \  and  his  wife  |.  It  happens  that  twins  are  born,  a 
boy  and  a  girl.  How  shall  the  estates  be  divided  so  as  to  satisfy  the 
will?  The  celebrated  Roman  jurist,  Salvianus  Julianus,  decided  that 
the  estates  shall  be  divided  into  seven  equal  parts  of  which  the  son 
receives  four,  the  wife  two,  the  daughter  one. 

We  next  consider  Roman  geometry.  He  who  expects  to  find  in 
^  Friedlein,  op.  cit.,  p.  89. 


66  A  HISTORY  OF  MATHEMATICS 

Rome  a  science  of  geometry,  with  definitions,  axioms,  theoiems,  and 
proofs  arranged  in  logical  order,  will  be  disappointed.  The  only 
geometry  known  was  a  practical  geometry,  which,  like  the  old  Egyp- 
tian, consisted  only  of  empirical  rules.  This  practical  geometry  was 
employed  in  surveying.  Treatises  thereon  have  come  down  to  us, 
compiled  by  the  Roman  surveyors,  called  agrimensores  or  gromatici. 
One  would  naturally  expect  rules  to  be  clearly  formulated.  But  no; 
they  are  left  to  be  abstracted  by  the  reader  from  a  mass  of  numerical 
examples.  "The  total  impression  is  as  though  the  Roman  gromatic 
were  thousands  of  years  older  than  Greek  geometry,  and  as  though 
a  deluge  were  lying  between  the  two."  Some  of  their  rules  were  prob- 
ably inherited  from  the  Etruscans,  but  others  are  identical  with  those 
of  Heron.  Among  the  latter  is  that  for  finding  the  area  of  a  triangle 
^om  its  sides  and  the  approximate  formula,  ^a-,  for  the  area  of 
equilateral  triangles  (a  being  one  of  the  sides).  But  the  latter  area 
was  also  calculated  by  the  formulas  \{a^+a)  and  \a^,  the  first  of 
which  was  unknown  to  Heron.  Probably  the  expression  |a2  was  de- 
rived from  the  Egyptian  formula . for  the  determination  of 

2  2 

the  surface  of  a  quadrilateral.  This  Egyptian  formula  was  used  by 
the  Romans  for  finding  the  area,  not  only  of  rectangles,  but  of  any 
quadrilaterals  whatever.  Indeed,  the  gromatici  considered  it  even 
sufficiently  accurate  to  determine  the  areas  of  cities,  laid  out  irregu- 
larly, simply  by  measuring  their  circumferences.  ^  Whatever  Egyptian 
geometry  the  Romans  possessed  was  transplanted  across  the  Mediter- 
ranean at  the  time  of  Julius  CcBsar,  who  ordered  a  survey  of  the  whole 
empire  to  secure  an  equitable  mode  of  taxation.  Casar  also  reformed 
the  calendar,  and,  for  that  purpose,  drew  from  Egyptian  learning. 
He  secured  the  services  of  the  Alexandrian  astronomer,  Sosigenes. 

Two  Roman  philosophical  writers  deserve  our  attention.  The 
philosophical  poet,  Titus  Lucretius  (96P-55  b.  c),  in  his  De  rerum 
natura,  entertains  conceptions  of  an  infinite  multitude  and  of  an  in- 
finite magnitude  which  accord  with  the  modern  definitions  of  those 
terms  as  being  not  variables  but  constants.  However,  the  Lucretian 
infinites  are  not  composed  of  abstract  things,  but  of  material  particles. 
His  infinite  multitude  is  of  the  denumerable  variety;  he  made  use 
of  the  whole-part  property  of  infinite  multitudes.^ 

Cognate  topics  are  discussed  several  centuries  later  by  the  cele- 
brated father  of  the  Latin  church,  St.  Augustine  (354-430  a.  d.),  in 
his  references  to  Zeno  of  Elea.  In  a  dialogue  on  the  question,  whether 
or  not  the  mind  of  man  moves  when  the  body  moves,  and  travels  with 
the  body,  he  is  led  to  a  definition  of  motion,  in  which  he  displays  some 
levity.    It  has  been  said  of  scholasticism  that  it  has  no  sense  of  humor. 

*  H.  Hankel,  op.  cit.,  p.  297. 

'  C.  J.  Keyset  in  Bull.  Am.  Math.  Soc,  Vol.  24,  1918,  p.  268,  321. 


THE  ROMANS  67 

Hardly  does  this  apply  to  St.  Augustine.  He  says:  "When  this  dis- 
course was  concluded,  a  boy  came  running  from  the  house  to  call  us 
to  dinner.  I  then  remarked  that  this  boy  compels  us  not  only  to 
define  motion,  but  to  see  it  before  our  very  eyes.  So  let  us  go,  and 
pass  from  this  place  to  another;  for  that  is,  if  I  am  not  mistaken, 
nothing  else  than  motion."  St.  Augustine  deserves  the  credit  of 
having  accepted  the  existence  of  the  actually  infinite  and  to  have 
recognized  it  as  being,  not  a  variable,  but  a  constant.  He  recognized 
all  finite  positive  integers  as  an  infinity  of  that  type.  On  this  point 
he  occupied  a  radically  different  position  than  his  forerunner,  the 
Greek  father  of  the  church,  Origen  of  Alexandria.  Origen's  arguments 
against  the  actually  infinite  have  been  pronounced  by  Georg  Cantor 
the  profoundest  ever  advanced  against  the  actually  infinite. 

In  the  fifth  century,  the  Western  Roman  Empire  was  fast  fallii^ 
to  pieces.  Three  great  branches — Spain,  Gaul,  and  the  province  of 
Africa — broke  off  from  the  decaying  trunk.  In  476,  the  Western 
Empire  passed  away,  and  the  Visigothic  chief,  Odoacer,  became  king. 
Soon  after,  Italy  was  conquered  by  the  Ostrogoths  under  Theodoric. 
It  is  remarkable  that  this  very  period  of  poUtical  humiliation  should 
be  the  one  during  which  Greek  science  was  studied  in  Italy  most 
zealously.  School-books  began  to  be  compiled  from  the  elements  of 
Greek  authors.  These  compilations  are  very  deficient,  but  are  of 
absorbing  interest,  from  the  fact  that,  down  to  the  twelfth  century, 
they  were  the  only  sources  of  mathematical  knowledge  in  the  Occident. 
Foremost  among  these  writers  is  Boethius  (died  524).  At  first  he 
was  a  great  favorite  of  King  Theodoric,  but  later,  being  charged  by 
envious  courtiers  with  treason,  he  was  imprisoned,  and  at  last  decapi- 
tated. While  in  prison  he  wrote  On  the  Consolations  of  Philosophy.  As 
a  mathematician,  Boethius  was  a  Brobdingnagian  among  Roman 
Scholars,  but  a  Liliputian  by  the  side  of  Greek  masters.  He  wrote 
an  Institutis  Arithmetica,  which  is  essentially  a  translation  of  the  arith- 
metic of  Nicomachus,  and  a  Geometry  in  several  books.  Some  of  the 
most  beautiful  results  of  Nicomachus  are  omitted  in  Boethius'  arith- 
metic. The  first  book  on  geometry  is  an  extract  from  Euclid's  Ele- 
ments, which  contains,  in  addition  to  definitions,  postulates,  and 
axioms,  the  theorems  in  the  first  three  books,  without  proofs.  How 
can  this  omission  of  proofs  be  accounted  for?  It  has  been  argued  by 
some  that  Boethius  possessed  an  incomplete  Greek  copy  of  the  Ele- 
ments; by  others,  that  he  had  Theon's  edition  before  him,  and  be- 
lieved that  only  the  theorems  came  from  Euclid,  while  the  proofs  were 
supplied  by  Theon.  The  second  book,  as  also  other  books  on  geometry 
attributed  to  Boethius,  teaches,  from  numerical  examples,  the  men- 
suration of  plane  figures  after  the  fashion  of  the  agrimensores. 

A  celebrated  portion  in  the  geometry  of  Boethius  is  that  pertaining 
to  an  abacus,  which  he  attributes  to  the  Pythagoreans.  A  consider- 
ible  improvement  on  the  old  abacus  is  there  introduced.     I'ebbles 


68  A  HISTORY  OF  MATHEMATICS 

are  discarded,  and  apices  (probably  small  cones)  are  used.  Upon  each 
of  these  apices  is  drawn  a  numeral  giving  it  some  value  below  lo. 
The  names  of  these  numerals  are  pure  Arabic,  or  nearly  so,  but  are 
added,  apparently,  by  a  later  hand.  The  o  is  not  mentioned  by 
Boethius  in  the  text.  These  numerals  bear  striking  resemblance  to 
the  Gubar-numerals  of  the  West-Arabs,  which  are  admittedly  of 
Indian  origin.  These  facts  have  given  rise  to  an  endless  controversy. 
Some  contended  that  Pythagoras  was  in  India,  and  from  there  brought 
the  nine  numerals  to  Greece,  where  the  Pythagoreans  used  them 
secretly.  This  hypothesis  has  been  generally  abandoned,  for  it  is 
not  certain  that  Pythagoras  or  any  disciple  of  his  ever  was  in  India, 
nor  is  there  any  evidence  in  any  Greek  author,  that  the  apices  were 
known  to  the  Greeks,  or  that  numeral  signs  of  any  sort  were  used  by 
them  with  the  abacus.  It  is  improbable,  moreover,  that  the  Indian 
signs,  from  which  the  apices  are  derived,  are  so  old  as  the  time  of 
Pythagoras.  A  second  theory  is  that  the  Geometry  attributed  to 
Boethius  is  a  forgery;  that  it  is  not  older  than  the  tenth,  or  possibly 
the  ninth,  century,  and  that  the  apices  are  derived  from  the  Arabs. 
But  there  is  an  Encyclopaedia  written  by  Cassiodorius  (died  about 
585)  in  which  both  the  arithmetic  and  geometry  of  Boethius  are  men- 
tioned. Some  doubt  exists  as  to  the  proper  interpretation  of  this 
passage  in  the  Encyclopaedia.  At  present  the  weight  of  evidence  is 
that  the  geometry  of  Boethius,  or  at  least  the  part  mentioning  the 
numerals,  is  spurious.i  A  third  theory  (Woepcke's)  is  that  the 
Alexandrians  either  directly  or  indirectly  obtained  the  nine  numerals 
from  the  Hindus,  about  the  second  century  a.  d.,  and  gave  them  to 
the  Romans  on  the  one  hand,  and  to  the  Western  Arabs  on  the  other. 
This  explanation  is  the  most  plausible. 

It  is  worthy  of  note  that  Cassiodorius  was  the  first  writer  to  use 
the  terms  "rational"  and  "irrational"  in  the  sense  now  current  in 
arithmetic  and  algebra.2 

^  A  good  discussion  of  this  so-called  "Boethius  question,"  which  has  been  de- 
bated for  two  centuries,  is  given  by  D.  E.  Smith  and  L.  C.  Karpinski  in  their  Hitidu- 
Arabic  Numerals,  191 1,  Chap.  V. 

2  Encyclopedic  des  sciences  mathemaliqiies,  Tome  I,  Vol.  2,  1907,  p.  2.  An  il- 
luminating article  on  ancient  finger-symbolism  is  L.  J.  Richardson's  "  Digital 
Reckoning  Among  the  Ancients"  in  the  Am.  Math.  Monthly,  Vol  22,,  1816, 
PP-  7-13- 


THE  MAYA 

The  Maya  of  Central  America  and  Southern  Mexico  developed 
hieroglyphic  writing,  as  found  in  inscriptions  and  codices  dating  ap- 
parently from  about  the  beginning  of  the  Christian  era,  that  ranks 
"probably  as  the  foremost  intellectual  achievement  of  pre-Columbian 
times  in  the  New  World."  Maya  number  systems  and  chronology 
are  remarkable  for  the  extent  of  their  early  development.  Perhaps 
five  or  six  centuries  before  the  Hindus  gave  a  systematic  exposition 
of  their  decimal  number  system  with  its  zero  and  principle  of  local 
value,  the  Maya  in  the  flatlands  of  Central  America  had  evolved 
systematically  a  vigesimal  number  system  employing  a  zero  and  the 
principle  of  local  value.  In  the  Maya  number  system  found  in  the 
codices  the  ratio  of  increase  of  successive  units  was  not  lo,  as  in  the 
Hindu  system;  it  was  20  in  all  positions  except  the  third.  That  is,. 
20  units  of  the  lowest  order  {kins,  or  days)  make  one  unit  of  the  next 
higher  order  {uinals,  or  20  days),  18  uinals  make  one  unit  of  the  third 
order  (tun,  or  300  aays),  20  tuns  make  one  unit  of  the  fourth  ordet 
(katmt,  or  7200  days),  20  katuns  make  one  unit  of  the  fifth  order 
(cycle,  or  144,000  days)  and  finally,  20  cycles  make  i  great  cycle  of 
2,880,000  days.  In  Maya  codices  we  find  symbols  for  i  to  19,  ex- 
pressed by  bars  and  dots.  Each  bar  stands  for  5  units,  each  dot  for 
I  unit.    For  instance, 


I        2         4       5       7         II  19 

The  zero  is  represented  by  a  symbol  that  looks  roughly  Hke  a  half- 
closed  eye.  In  writing  20  the  principle  of  local  value  enters.  It  is 
expressed  by  a  dot  placed  over  the  symbol  for  zero.  The  numbers 
are  written  vertically,  the  lowest  order  being  assigned  the  lowest 
position.  Accordingly,  37  was  expressed  by  the  symbols  for  17  (three 
bars  and  two  dots)  in  the  kin  place,  and  one  dot  representing  20, 
placed  above  17  in  the  uinal  place.  To  write  360  the  Maya  drew 
two  zeros,  one  above  the  other,  with  one  dot  higher  up,  in  third  place 
(1X18x20+0-1-0=360).  The  highest  number  found  in  the  codices 
is  in  our  decimal  notation  12,489,781. 

A  second  numeral  system  is  found  on  Maya  inscriptions.  It  em- 
ploys the  zero,  but  not  the  principle  of  local  value.  Special  symbols 
are  employed  to  designate  the  different  units.  It  is  as  if  we  were  to 
write  203  as  "2  hundreds,  o  tens,  3  ones,"  ^ 

'  For  an  account  of  the  Maya  number-systems  and  chronology,  see  S.  G.  Morley 
A II  Introduction  to  the  Study  of  the  Maya  Hierogliphs,  Government  Printing  Office 
^Vashinglon,  1915. 

)0 


70  A  HISTORY  OF  MATHEMATICS 

The  Maya  had  a  sacred  year  of  260  days,  an  ofl5cial  year  of  360 
days  and  a  solar  year  of  365+  days.  The  fact  that  18x20=360 
seems  to  account  for  the  break  in  the  vigesimal  system,  making  18 
(instead  of  20)  uinals  equal  to  i  tun.  The  lowest  common  multiple 
of  260  and  365,  or  18980,  was  taken  by  the  Maya  as  the  "calendar 
round,"  a  period  of  52  years,  which  is  "the  most  important  period  in 
Maya  chronology." 

We  may  add  here  that  the  number  systems  of  Indian  tribes  in  North 
America,  while  disclosing  no  use  of  the  zero  nor  of  the  principle  of 
local  value,  are  of  interest  as  exhibiting  not  only  quinary,  decimal,  and 
vigesimal  systems,  but  also  ternary,  quarternary,  and  octonary  sys- 
tems.^ 

1  See  W.  C.  Eells,  "Number  Systems  of  the  North  American  Indians"  in  Amer- 
ican Math.  Monthly,  Vol.  20,  i9'i3,  pp.  263-272,  293-299;  also  BibliotJieca  malhe- 
malica,  3  S.,  Vol.  13,  1913,  pp.  218-222. 


THE  CHINESE  i 

The  oldest  extant  Chinese  work  of  mathematical  interest  is  an 
anonymous  publication,  called  Chou-pei  and  written  before  the 
second  century,  a.  d.,  perhaps  long  before.  In  one  of  the  dialogues  the 
Chou-pei  is  believed  to  reveal  the  state  of  mathematics  and  astronomy 
in  China  as  early  as  iioo  b.  c.  The  Pythagorean  theorem  of  the  right 
triangle  appears  to  have  been  known  at  that  early  date. 

Next  to  the  Chou-pei  in  age  is  the  Chiu-chang  Suan-shu  ("Arith- 
metic in  Nine  Sections"),  commonly  called  the  Chiu-chang,  the  most 
celebrated  Chinese  Text  on  arithmetic.  Neither  its  authorship  nor 
the  time  of  its  composition  is  known  definitely.  By  an  edict  of  the 
despotic  emperor  Shih  Hoang-ti  of  the  Ch'in  Dynasty  "all  books  were 
burned  and  all  scholars  were  buried  in  the  year  213  b.  c."  After  the 
death  of  this  emperor,  learning  revived  again.  We  are  told  that  a 
scholar  named  Chang  T'sang  found  some  old  writings,  upon  which 
he  based  this  famous  treatise,  the  Chiu-chang.  About  a  century  later 
a  revision  of  it  was  made  by  Ching  Ch'ou-ch'ang;  commentaries  on 
this  classic  text  were  made  by  Liu  Hui  in  263  a.  d.  and  by  Li  Ch'un- 
feng  in  the  seventh  century.  How  much  of  the  "Arithmetic  in  Nine 
Sections,"  as  it  exists  to-day,  is  due  to  the  old  records  ante-dating 
213  B.  c,  how  much  to  Chang  T'sang  and  how  much  to  Ching  Ch'ou- 
ch'ang,  it  has  not  yet  been  found  possible  to  determine. 

The  "Arithmetic  in  Nine  Sections"  begins  with  mensuration;  it 
gives  the  area  of  a  triangle  as  |  6  h,  of  a  trapezoid  as  |  {b  +b')h,  of  a 
circle  variously  as  |c  .^t/,  led,  |i-  and  -^^c-,  where  c  is  the  circumference 
and  d  is  the  diameter.  Here  tt  is  taken  equal  to  3.  The  area  of  a 
segment  of  a  circle  is  given  as  ^(ca+a-),  where  c  is  the  chord  and  a 
the  altitude.  Then  follow  fractions,  commercial  arithmetic  including 
percentage  and  proportion,  partnership,  and  square  and  cube  root  of 
numbers.  Certain  parts  exhibit  a  partiality  for  unit-fractions.  Divi- 
sion by  a  fraction  is  effected  by  inverting  the  fraction  and  multiplying. 
The  rules  of  operation  are  usually  stated  in  obscure  language.  There 
are  given  rules  for  finding  the  volumes  of  the  prism,  cylinder,  pyramid, 
truncated  pyramid  and  cone,  tetrahedron  and  wedge.  Then  follow 
problems  in  alligation.  There  are  indications  of  the  use  of  positive 
and  negative  numbers.  Of  interest  is  the  following  problem  because 
centuries  later  it  is  found  in  a  work  of  the  Hindu  Brahmagupta: 

'  All  our  information  on  Chinese  mathematics  is  drawn  from  Yoshio  Mikami's 
The  Development  of  Mathematics  in  China  and  Japan,  Leipzig,  19 12,  and  from  David 
Eugene  Smith  and  Yoshio  Mikami's  History  of  Japanese  Mathematics,  Chicago, 
1914. 

71 


72  A  HISTORY  OF  MATHEMATICS 

There  is  a  bamboo  lo  ft.  high,  the  upper  end  of  which  is  broken  and 
reaches  to  the  ground  3  ft.  from  the  stem.    What  is  the  height  of  the 

break?    In  the  solution  the  height  of  the  break  is  taken  =Y-  t^ttt' 

2X  10 

Here  is  another:  A  square  town  has  a  gate  at  the  mid-point  of  each 
side.  Twenty  paces  north  of  the  north  gate  there  is  a  tree  which 
is  visible  from  a  point  reached  by  walking  from  the  south  gate  14 
paces  south  and  then  1775  paces  west.  Find  the  side  of  the  square. 
The  problem  leads  to  the  quadratic  equation  a:2+(2o  +  i4)x— 2X10X 
jy^^  ^Q_  The  derivation  and  solution  of  this  equation  are  not  made 
clear  in  the  text.  There  is  an  obscure  statement  to  the  effect  that 
the  answer  is  obtained  by  evolving  the  root  of  an  expression  which 
is  not  monomial  but  has  an  additional  term  [the  term  of  the  first 
degree  (20  +  i4)x].  It  has  been  surmised  that  the  process  here  re- 
ferred to  was  evolved  more  fully  later  and  led  to  the  method  closely 
resembling  Horner's  process  of  approximating  to  the  roots,  and  that 
the  process  was  carried  out  by  the  use  of  calculating  boards.  Another 
problem  leads  to  a  quadratic  equation,  the  rule  for  the  solution  of 
which  fits  the  solution  of  Uteral  quadratic  equations. 

We  come  next  to  the  Sun-Tsu  Suan-ching  ("Arithmetical  Classic 
of  Sun-Tsu"),  which  belongs  to  the  first  century,  a.  d.  The  author, 
SuN-Tsu,  says:  "In  making  calculations  we  must  first  know  positions 
of  numbers.  Unity  is  vertical  and  ten  horizontal;  the  hundred  stands 
while  the  thousand  lies;  and  the  thousand  and  the  ten  look  equally, 
and  so  also  the  ten  thousand  and  the  hundred."  This  is  evidently  a 
reference  to  abacal  computation,  practiced  from  time  immemorial  in 
China,  and  carried  on  by  the  use  of  computing  rods.  These  rods, 
made  of  small  bamboo  or  of  wood,  were  in  Sun-Tsu's  time  much  longer. 
The  later  rods  were  about  i|  inches  long,  red  and  black  in  color, 
representing  respectively  positive  and  negative  numbers.  According 
to  Sun-Tsu,  units  are  represented  by  vertical  rods,  tens  by  horizontal 
rods,  hundreds  by  vertical,  and  so  on;  for  5  a  single  rod  suffices.  The 
numbers  1-9  are  represented  by  rods  thus:  I,  II,  111,  nil,  mil,  |,J|,  1||,  ||||; 
the  numbers  in  the  tens  column,  10,  20,  .  .  .,  90  are  written  thus: 

_  =^  =^  =,  ^,  _1_,  J_,  =,  =.  The  number  6728  is  designated 
by  I  IT  ^^  TlT-  ^^^  ^°^^  ^^^^  placed  on  a  board  ruled  in  columns, 
and'were  rearranged  as  the  computation  advanced.  The  successive 
steps  in  the  multiplication  of  321  by  46  must  have  been  about  as 
follows: 

321  321  321 

138  1472  14766 

46  46  46 

The  product  was  placed  between  the  multiplicand  and  multiplier. 

The  46  is  multiplied  first  by  3,  then  by  2,  and  last  by  i,  the  46  being 


THE  CHINESE  73 

moved  to  the  right  one  place  at  each  step.  Sun-Tsu  does  not  take  up 
divasion,  except  when  the  divisor  consists  of  one  digit.  Square  root 
is  explained  more  clearly  than  in  the  "Arithmetic  in  Nine  Sections." 
Algebra  is  involved  in  the  problem  suggested  by  the  reply  made  by  a 
woman  washing  dishes  at  a  river:  "I  don't  know  how  many  guests 
there  were;  but  every  two  used  a  dish  for  rice  between  them;  every 
three  a  dish  for  broth;  every  four  a  dish  for  meat;  and  there  were  65 
dishes  in  all. — Rule:  Arrange  the  65  dishes,  and  multiply  by  12,  when 
we  get  780.    Divide  by  13,  and  thus  we  obtain  the  answer." 

An  indeterminate  equation  is  involved  in  the  following:  "There  are 
certain  things  whose  number  is  unknown.  Repeatedly  divide  by  3, 
the  remainder  is  2 ;  by  5  the  remainder  is  3 ;  and  by  7  the  remainder  is 
2.    What  will  be  the  number?"    Only  one  solution  is  given,  viz.  23. 

The  Hai-tao  Suan-ching  ("Sea-island  Arithmetical  Classic")  was 
written  by  Liu  Hui,  the  commentator  on  the  "Arithmetic  in  Nine 
Sections,"  during  the  war-period  in  the  third  century,  a.  d.  He  gives 
complicated  problems  indicating  marked  proficiency  in  algebraic 
manipulation.  The  first  problem  calls  for  the  determination  of  the 
distance  of  an  island  and  the  height  of  a  peak  on  the  island,  when  two 
rods  30'  high  and  1000'  apart  are  in  line  with  the  peak,  the  top  of  the 
peak  being  in  line  with  the  top  of  the  nearer  (more  remote)  rod,  when 
seen  from  a  point  on  the  level  ground  123'  (127')  behind  this  nearer 
(more  remote)  rod.  The  rules  given  for  solving  the  problem  are 
equivalent  to  the  expressions  obtained  from  proportions  arising  from 
the  similar  triangles. 

Of  the  treatises  brought  forth  during  the  next  centuries  only  a  few 
are  extant.  We  mention  the  "Arithmetical  Classic  of  Chang  Ch'iu- 
chien"  of  the  sbcth  century  which  gives  problems  on  proportion,  arith- 
metical progression  and  mensuration.  He  proposes  the  "problem  of 
100  hens"  which  is  given  again  by  later  Chinese  authors:  "A  cock 
costs  5  pieces  of  money,  a  hen  3  pieces,  and  3  chickens  i  piece.  If 
then  we  buy  with  100  pieces  100  of  them,  what  will  be  their  respective 
numbers?" 

The  early  values  of  ir  used  in  China  were  3  and  vio^  i-iu  Hui 
calculated  the  perimeters  of  regular  inscribed  polygons  of  12,  24,  48, 
96,  192  sides  and  arrived  at  7r  =3.14+.  Tsu  Ch'ung-chih  in  the  fifth 
century  took  the  diameter  10^  and  obtained  as  upper  and  lower  limits 
for  TT  3.1415927  and  3.1415926,  and  from  these  the  "accurate"  and 
"inaccurate"  values  355/113,  22/7.  The  value  22/7  is  the  upper  Umit 
given  by  Archimedes  and  is  found  here  for  the  first  time  in  Chinese 
history.  The  ratio  355/113  became  known  to  the  Japanese,  but  in 
the  West  it  was  not  known  until  Adriaen  Anthonisz,  the  father  of 
Adriaen  Melius,  derived  it  anew,  sometime  between  1585  and  1625. 
However,  M.  Curtze's  researches  would  seem  to  show  that  it  was 
known  to  Valentin  Otto  as  early  as  1573.^ 

1  Bibliotheca  malhemalica,  3  S.,  Vol.  13,  19 13,  p.  264.    A  neat  geometric  construe- 


74  A  HISTORY  OF  MATHEMATICS 

In  the  first  half  of  the  seventh  century  Wang  Hs'  iao-t'ung  brought 
forth  a  work,  the  Ch'i-ku  Siian-ching,  in  which  numerical  cubic  equa- 
tions appear  for  the  first  time  in  Chinese  mathematics.  This  took 
place  seven  or  eight  centuries  after  the  first  Chinese  treatment  of 
quadratics.  Wang  Hs'iao-t'ung  gives  several  problems  leading  to 
cubics:  "There  is  a  right  triangle,  the  product  of  whose  two  sides  is 
706  -gQ, and  whose  hypotenuse  is  greater  than  the  first  side  by  30  g"^. 
It  is  required  to  know  the  lengths  of  the  three  sides."  He  gives  the 
answer  14  -^,  49  \,  S'^-h  ^^^  the  rule:  "The  Product  (P)  being 
squared  and  being  divided  by  twice  the  Surplus  (S),  make  the  result 
shih  or  the  constant  class.  Halve  the  surplus  and  make  it  the  lien-fa 
or  the  second  degree  class.  And  carry  out  the  operation  of  evolution 
according  to  the  extraction  of  cube  root.  The  result  gives  the  first 
side.  Adding  the  surplus  to  it,  one  gets  the  hypotenuse.  Divide  the 
product  with  the  first  side  and  the  quotient  is  the  second  side."  This 
rule  leads  to  the  cubic  equation  x^  +S/2.v^-|'^  =0,  The  mode  of  solu- 
tion is  similar  to  the  process  of  extracting  cube  roots,  but  details  of 
the  process  are  not  revealed. 

In  1247  Ch'in  Chiu-shao  wrote  the  Su-shu  Chiu-chang  ("Nine 
Sections  of  Mathematics")  which  makes  a  decided  advance  on  the 
solution  of  numerical  equations.  At  first  Ch'in  Chiu-shao  led  a  mili- 
tary life;  he  lived  at  the  time  of  the  Mongolian  invasion.  For  ten 
years  stricken  with  disease,  he  recovered  and  then  devoted  himself  to 
study.  The  following  problem  led  him  to  an  equation  of  the  tenth 
degree:  There  is  a  circular  castle  of  unknown  diameter,  having  4 
gates.  Three  miles  north  of  the  north  gate  is  a  tree  which  is  visible 
from  a  point  9  miles  east  of  the  south  gate.  The  unknown  diameter 
is  found  to  be  9.  He  passes  beyond  Sun-Tsu  in  his  ability  to  solve 
indeterminate  equations  arising  for  a  number  which  will  give  the 
residues  ri,  r2,  .  .,  r^  when  divided  by  nii,  m<i,  .,  Wn,  respectively. 

Ch'in  Chiu-shao  solves  the  equation  -x*-|-7632oox-  — 40642560000 
=  0  by  a  process  almost  identical  with  Horner's  method.  However, 
the  computations  were  very  probably  carried  out  on  a  computing 
board,  divided  into  columns,  and  by  the  use  of  computing  rods. 
Hence  the  arrangement  of  the  work  must  have  been  different  from 
that  of  Horner.  But  the  operations  performed  were  the  same.  The 
first  digit  in  the  root  being  8,  (8  hundreds),  a  transformation  is  ef- 
fected which  yields  .r^  — 3  2oox^  —  30768oox'-^- 826880000.x -I-3820544- 
0000=0,  the  same  equation  that  is  obtained  by  Horner's  process. 
Then,  taking  4  as  the  second  figure  in  the  root,  the  absolute  term 
vanishes  in  the  operation,  giving  the  root  840.    Thus  the  Chinese  had 

tion  of  the  fraction  ^  '^  |  =3  +4^  -^  (7^+8-)  is  given  anonymously  in  Grunert's  Archiv 
Vol.  12,  1849,  p.  98.  Using  ^'Jl,  T.  M.  P.  Hughes  gives  in  Nature,  Vol.  93, 
T914,  p.  no,  a  method  of  constructing  a  triangle  that  gives  the  area  of  a  given 
circle  with  great  accuracy. 


THE  CHINESE 


75 


invented  Homer's  method  of  solving  numerical  equations  more  than 
five  centuries  before  Ruffini  and  Horner.  This  solution  of  higher 
numerical  equations  is  given  later  in  the  writings  of  Li  Yeh  and  others. 
Ch'in  Chiu-shao  marks  an  advance  over  Sun-Tsu  in  the  use  of  o  as  a 
symbol  for  zero.  Most  likely  this  symbol  is  an  importation  from 
India.  Positive  and  negative  numbers  were  distinguished  by  the  use 
of  red  and  black  computing  rods.  This  author  gives  for  the  first  time 
a  problem  which  later  became  a  favorite  one  among  the  Chinese;  it 
involved  the  trisection  of  a  trapezoidal  field  under  certain  restrictions 
in  the  mode  of  selection  of  boundaries. 

We  have  already  mentioned  a  contemporary  of  Ch'in  Chiu-shao, 
namely,  Li  Yeh;  he  lived  far  apart  in  a  rival  monarchy  and  worked 
independently.  He  was  the  author  of  Tse-yuan  Hai-ching  ("Sea- 
Mirror  of  the  Circle-Measurements"),  1248,  and  of  the  I-ku  Ven-liian, 
1259.  He  used  the  symbol  o  for  zero.  On  account  of  the  inconven- 
ience of  writing  and  printing  positive  and  negative  numbers  in  dif- 
ferent colors,  he  designated  negative  numbers  by  drawing  a  cancella- 
tion mark  across  the  symbol.  Thus  J_o  stood  for  60,  JKo  stood  for 
-  60.  The  unknown  quantity  was  represented  by  unity  which  was 
probably  represented  on  the  counting  board  by  a  rod  easily  distin- 
guished from  the  other  rods.  The  terms  of  an  equation  were  written, 
not  in  a  horizontal,  but  in  a  vertical  line.  In  Li  Yeh's  work  of  1259, 
as  also  in  the  work  of  Ch'in  Chiu-shao,  the  absolute  term  is  put  in  the 
top  line;  in  Li  Yeh's  work  of  1248  the  order  of  the  terms  is  reversed, 
so  that  the  absolute  term  is  in  the  bottom  line  and  the  highest  power 
of  the  unknown  in  the  top  line.  In  the  thirteenth  century  Chinese 
algebra  reached  a  much  higher  development  than  formerly.  This 
science,  with  its  remarkable  method  (our  Horner's)  of  solving  numer- 
ical equations,  was  designated  by  the  Chinese  "the  celestial  element 
method." 

A  third  prominent  thirteenth  century  mathematician  was  Yang 
Hui,  of  whom  several  books  are  still  extant.  They  deal  with  the 
summation  of  arithmetical  progressions,  of  the  series  1+^+6  +  .  .  + 
(1-I-2-I-.  .  +n)  =n{ji  +  i}{n+2)^6,  12+22-f  .  .  +^2  =i?i(«+i)(»+i), 
also  with  proportion,  simultaneous  linear  equations,  quadratic  and 
quartic  equations. 

Half  a  century  later,  Chinese  algebra  reached  its  height  in  the 
treatise  Suan-hsiao  Chi-meng  ("Introduction  to  Mathematical 
Studies"),  1299,  and  the  Szu-yiien  Vu-chien  ("The  Precious  Mirror 
of  the  Four  Elements"),  1303,  which  came  from  the  pen  of  Chu 
Shih-Chieh.  The  first  work  contains  no  new  results,  but  exerted  a 
great  stimulus  on  Japanese  mathematics  in  the  seventeenth  century. 
At  one  time  the  book  was  lost  in  China,  but  in  1839  it  was  restored 
by  the  discovery  of  a  copy  of  a  Korean  reprint,  made  in  lOOo.  The 
'Vrecious  Mirror"  is  a  more  original  work.  It  treats  fully  of  the 
"celestial  element  method."     He  gives  as  an  "ancient  method"  a 


76 


A  HISTORY  OF  MATHEMATICS 


triangle  (known  in  the  West  as  Pascal's  arithmetical  triangle),  dis- 
playing the  binomial  coefficients,  which  were  known  to  the  Arabs  in 
the  eleventh  century  and  were  probably  imported  into  China.  Chu 
shih-Chieh's  algebraic  notation  was  altogether  different  from  our 
modern  notation.    Thus,  a  +o  +c  +d  was  written 


I 
1*1 

I 


I 
202 

I    O   *2  O    I 

22  O    2 


as  shown  on  the  left,  except  that,  in  the  central  position,  we  employ 
an  asterisk  in  place  of  the  Chinese  character  t'ai  (great  extreme,  ab- 
solute term)  and  that  we  use  the  modern  numerals  in  place  of  the 
sangi  forms.  The  square  of  a  -\-b  +c  -f-J,  namely,  a-  -\-b'^  +c^  +d^  +  2ab 
+2ac+2ad+2bc+2bd+2cd,  is  represented  as  shown  on  the  right. 
In  further  illustration  of  the  Chinese  notation,  at  the  time  of  Chu 
Shih-Chieh,  we  give  1 


' 

•  1  -■■ 

* 

hi- 

* 

-2 

.-.= 

, 

0 

'1 

I 

* 

* 

0 

02 

* 



=  11 



=  .v 





=  xz 





=  2 

I 

0 

I 

0 

0 

In  the  fourteenth  century  astronomy  and  the  calendar  were  studied 
They  involved  the  rudiments  of  geometry  and  spherical  trigonometry. 
In  this  field  importations  from  the  Arabs  are  disclosed. 

After  the  noteworthy  achievements  of  the  thirteenth  century, 
Chinese  mathematics  for  several  centuries  was  in  a  period  of  decline. 
The  famous  "celestial  element  method"  in  the  solution  of  higher 
equations  was  abandoned  and  forgotten.  Mention  must  be  made, 
however,  of  Ch'eng  Tai-wei,  who  in  1593  issued  his  Suan-fa  T'ung- 
tsung  ("A  Systematised  Treatise  on  Arithmetic"),  which  is  the  oldest 
work  now  extant  that  contains  a  diagram  of  the  form  of  the  abacus, 
called  suan-pan,  and  the  explanation  of  its  use.  The  instrument  was 
known  in  China  in  the  twelfth  century.  Resembling  the  old  Roman 
abacus,  it  contained  balls,  movable  along  rods  held  by  a  wooden 
frame.  The  suan-pan  replaced  the  old  computing  rods.  The  "Sys- 
tematised Treatise  on  Arithmetic"  is  famous  also  for  containing  some 
magic  squares  and  magic  circles.    Little  is  known  of  the  early  history 

^In  the  symbol  for  "xz"  notice  that  the  "i"  is  one  space  down  (x)  and  one 
space  to  the  right  (2)  of  *,  and  is  made  to  stand  for  the  product  xz.  In  the  symbol 
for  "2y3"  the  three  o's  indicate  the  absence  of  the  terms  y,  x,  xy;  the  small  "2" 
means  twice  the  product  of  the  two  letters  in  the  same  row,  respectively  one  space 
to  the  right  and  to  the  left  of  *,  i.  e.,  2  yz.  The  limitations  of  this  notation  are  ob- 
vious. 


THE  CHINESE 


77 


of  magic  squares.  Myth  tells  us  that,  in  early  times,  the  sage  Yii, 
the  enlightened  emperor,  saw  on  the  calamitous  Yellow  River  a  divine 
tortoise,  whose  back  was  decorated  with  the  figure  made  up  of  the 
numbers  from  i  to  9,  arranged  in  form  of  a  magic  square  or  lo-shu. 


\^\  o  00000000  / 


4 

9 

5 

I 

2 

7 
6 

3 
8 

^  '  <> 


The  lo-shu. 

The  numerals  are  indicated  by  knots  in  strings:  black  knots  repre- 
sent even  numbers  (symbolizing  imperfection),  white  knots  repre- 
sent odd  numbers  (perfection). 

Christian  missionaries  entered  China  in  the  sixteenth  century. 
The  Italian  Jesuit  Matteo  Ricci  (1552-1610)  introduced  European 
astronomy  and  mathematics.  With  the  aid  of  a  Chinese  scholar 
named  Hsu,  he  brought  out  in  1607  a  translation  of  the  first  six  books 
of  EucHd.  Soon  after  followed  a  sequel  to  Euclid  and  a  treatise  on 
surveying.  The  missionary  Mu  Ni-ko  sometime  before  1660  intro- 
duced logarithms.  In  17 13  Adrian  Vlack's  logarithmic  tables  to  11 
places  were  reprinted.  Ferdinand  Verbiest  ^  of  West  Flanders,  a 
noted  Jesuit  missionary  and  astronomer,  was  in  1669  made  vice- 
president  of  the  Chinese  astronomical  board  and  in  1673  its  president. 
European  algebra  found  its  way  into  China.  Mei  Ku-ch'ing  noticed 
that  the  European  algebra  was  essentially  of  the  same  principles  as 
the  Chinese  "celestial  element  method"  of  former  days  which  had 
been  forgotten.  Through  him  there  came  a  revival  of  their  own 
algebraic  method,  without,  however,  displacing  European  science. 
Later  Chinese  studies  touched  mainly  three  subjects:  The  determina- 
tion of  TT  by  geometry  and  by  infinite  series,  the  solution  of  numerical 
equations,  and  the  theory  of  logarithms. 

We  shall  see  later  that  Chinese  mathematics  stimulated  the  growth 
of  mathematics  in  Japan  and  India.  We  have  seen  that,  in  a  small 
way,  there  was  a  taking  as  well  as  a  giving.  Before  the  influx  of 
recent  European  science,  China  was  influenced  somewhat  by  Hindu 
and  Arabic  mathematics.  The  Chinese  achievements  which  stand 
out  most  conspicuously  are  the  solution  of  numerical  equations  and 
the  origination  of  magic  squares  and  magic  circles. 

^  Consult  H.  Bosnians,  Ferdinand  Verbiest,  Louvain,  191 2.  Extract  from  Revue 
des  Questions  scienlifiqiics,  January-April,  1912. 


THE  JAPANESE  * 

According  to  tradition,  there  existed  in  Japan  in  remote  times  a 
system  of  numeration  which  extended  to  high  powers  of  ten  and  re- 
sembled somewhat  the  sand  counter  of  Archimedes.  About  552  a.  d. 
Buddhism  was  introduced  into  Japan.  This  new  movement  was 
fostered  by  Prince  Shotoku  Tatshi  who  was  deeply  interested  in  all 
learning.  Mathematics  engaged  his  attention  to  such  a  degree  that 
he  came  to  be  called  the  father  of  Japanese  mathematics.  A  little 
later  the  Chinese  system  of  weights  and  measures  was  adopted.  In 
701  a  university  system  was  established  in  which  mathematics  figured 
prominently.  Chinese  science  was  imported,  special  mention  being 
made  in  the  official  Japanese  records  of  nine  Chinese  texts  on  mathe- 
matics, which  include  the  Chou-pei,  the  Suan-ching  written  by  Sun- 
Tsu  and  the  great  arithmetical  work,  the  Chiu-chang.  But  this  eighth 
century  interest  in  mathematics  was  of  short  duration;  the  Chiu-chang 
was  forgotten  and  the  dark  ages  returned.  Calendar  reckoning  and 
the  rudiments  of  computation  are  the  only  signs  of  mathematical 
activity  until  about  the  seventeenth  century  of  our  era.  On  account 
of  the  crude  numeral  systems,  devoid  of  the  principal  of  local  value 
and  of  a  symbol  for  zero,  mechanical  aids  of  computation  became  a 
necessity.  These  consisted  in  Japan,  as  in  China,  of  some  forms  of 
the  abacus.  In  China  there  came  to  be  developed  an  instrument, 
called  the  suan-pan,  in  Japan  it  was  called  the  soroban.  The  importa- 
tion of  the  suan-pan  into  Japan  is  usually  supposed  to  have  occurred 
before  the  close  of  the  sixteenth  century.  Bamboo  computing  rods 
were  used  in  Japan  in  the  seventh  century.  These  round  pieces  were 
replaced  later  by  the  square  prisms  (sangi  pieces).  Numbers  were 
represented  by  these  rods  in  the  manner  practiced  by  the  Chinese. 
The  numerals  were  placed  inside  the  squares  of  a  surface  ruled  like  a 
chess  board.  The  sorohan  was  simply  a  more  highly  developed  form 
of  abacal  instrument. 

The  years  1600  to  1675  mark  a  period  of  great  mathematical  ac- 
tivity. It  was  inaugurated  by  Mori  Kambei  Shigeyoshi,  who  popu- 
larized the  use  of  the  sorohan.  His  pupil,  Yoshida  Shichibei  Koyu, 
is  the  author  of  Jink'o-ki,  1627,  which  attained  wide  popularity  and 
is  the  oldest  Japanese  mathematical  work  now  extant.  It  explains 
operations  on  the  soroban,  including  square  and  cube  root.    In  one  of 

^This  account  is  compiled  from  David  Eugene  Smith  and  "S^oshio  Mikami's 
History  of  Japanese  Mathematics,  Chicago,  19 14,  from  Yoshio  Mikami's  Develop- 
ment of  Mathematics  in  China  and  Japan,  Leipzig,  191 2,  and  from  T.  Hayashi's 
A  Brief  History  of  the  Japanese  Mathematics,  Overgedrukt  uit  het  Nieuvj  Archiej 
voor  Wiskunde  VI,  pp.  296-361;  VII,  pp.  105-161. 

78 


THE  JAPANESE  79 

his  later  editions  Yoshida  appended  a  number  of  advanced  problems 
to  be  solved  by  competitors.  This  procedure  started  among  the 
Japanese  the  practice  of  issuing  problems,  which  was  kept  up  until 
1813  and  helped  to  stimulate  mathematical  activity. 

Another  pupil  of  Mori  was  Imamura  Chisho  who,  in  1639,  pub- 
lished a  treatise  entitled  Jugairoku,  written  in  classical  Chinese.  He 
took  up  the  mensuration  of  the  circle,  sphere  and  cone.  Another 
author,  Isomura  Kittoku,  in  his  Ketsugisho,  1660  (second  edition 
1684),  when  considering  problems  on  mensuration,  makes  a  crude 
approach  to  integration.  He  gives  magic  squares,  both  odd  and  even 
celled,  and  also  magic  circles.  Such  squares  and  circles  became  favor- 
ite topics  among  the  Japanese.  In  the  1684  edition,  Isomura  gives 
also  magic  wheels.  Tanaka  Kisshin  arranges  the  integers  1-96  in 
six  4--celled  magic  squares,  such  that  the  sum  in  each  row  and  column 
are  194;  placing  the  six  squares  upon  a  cube,  he  obtains  his  "magic 
cube."  Tanaka  formed  also  "magic  rectangles."  ^  Muramatsu  in 
1663  gives  a  magic  square  containing  as  many  as  19^  cells  and  a  magic 
circle  involving  129  numbers.  Muramatsu  gives  also  the  famous 
"  Josephus  Problem"  in  the  following  form:  'Once  upon  a  time  there 
lived  a  wealthy  farmer  who  had  thirty  children,  half  being  of  his  first 
wife  and  half  of  his  second  one.  The  latter  wished  a  favorite  son  to 
inherit  all  the  property,  and  accordingly  she  asked  him  one  day,  say- 
ing: Would  it  not  be  well  to  arrange  our  30  children  on  a  circle,  calling 
one  of  them  the  first  and  counting  out  every  tenth  one  until  there 
should  remain  only  one,  who  should  be  called  the  heir.  The  hus- 
band assenting,  the  wife  arranged  the  children  .  .  ;  the  counting  .  . 
resulted  in  the  elimination  of  14  step-children  at  once,  leaving  only 
one.  Thereupon  the  wife,  feeling  confident  of  her  success,  said,  .  , 
let  us  reverse  the  order.  .  The  husband  agreed  again,  and  the 
counting  proceeded  in  the  reverse  order,  with  the  unexpected  result 
that  all  of  the  second  wife's  children  were  stricken  out  and  there  re- 
mained only  the  step-child,  and  accordingly  he  inherited  the  property." 
The  origin  of  this  problem  is  not  known.  It  is  found  much  earUer  in 
the  Codex  Einsidelensis  (Einsideln,  Switzerland)  of  the  tenth  century, 
while  a  Latin  work  of  Roman  times  attributes  it  to  Flavius  Josephus. 
It  commonly  appears  as  a  problem  relating  to  Turks  and  Christians, 
half  of  whom  must  be  sacrificed  to  save  a  sinking  ship.  It  was  very 
common  in  early  printed  European  books  on  arithmetic  and  in  books 
on  mathematical  recreations. 

In  1666  Sato  Seik5  wrote  his  Kongenki  which,  in  common  with 
other  works  of  his  day,  considers  the  computation  of  7r(=3.i4). 
He  is  the  first  Japanese  to  take  up  the  Chinese  "celestial  element 
method"  in  algebra.  He  applies  it  to  equations  of  as  high  a  degree  as 
the  sbcth.  His  successor,  Sawaguciii,  and  a  contemporary  Nozawa, 
give  a  crude  calculus  resembling  that  of  Cavalieri.  Sawaguchi  rises 
1 Y.  Mikami  in  Archiv  der  Malhematik  u.  Pkysik,  Vol.  ^o,  pp.  183-186. 


8o  A  HISTORY  OF  MATHEMATICS 

above  the  Chinese  masters  in  recognising  the  plurality  of  roots,  but 
he  declares  problems  which  yield  them  to  be  erroneous  in  their  nature. 
Another  evidence  of  a  continued  Chinese  influence  is  seen  in  the 
Chinese  value  of  tt,  f  |^|,  which  was  made  known  in  Japan  by  Ikeda. 

We  come  now  to  Seki  Kowa  (1642-1708)  whom  the  Japanese  con- 
sider the  greatest  mathematician  that  their  country  has  produced. 
The  year  of  his  birth  was  the  year  in  which  Galileo  died  and  Newton 
was  born.  Seki  was  a  great  teacher  who  attracted  many  gifted  pupils. 
Like  Pythagoras,  he  discouraged  divulgence  of  mathematical  dis- 
coveries made  by  himself  and  his  school.  For  that  reason  it  is  difficult 
to  determine  with  certainty  the  exact  origin  and  nature  of  some  of  the 
discoveries  attributed  to  him.  He  is  said  to  have  left  hundreds  of 
manuscripts;  the  transcripts  of  a  few  of  them  still  remain.  He  pub- 
lished only  one  book,  the  Hatsuhi  Samp'o,  1674,  in  which  he  solved 
15  problems  issued  by  a  contemporary  writer.  Seki's  explanations 
are  quite  incomplete  and  obscure.  Takebe,  one  of  his  pupils,  lays 
stress  upon  Seki's  clearness.  The  inference  is  that  Seki  gave  his  ex- 
planations orally,  probably  using  the  computing  rods  or  sangi,  as  he 
proceeded.  Noteworthy  among  his  mathematical  achievements  are 
the  tenzan  method  and  the  yendan  method.  Both  of  these  refer  to 
improvements  in  algebra.  The  tenzan  method  is  an  improvement  of 
the  Chinese  "celestial  element"  method,  and  has  reference  to  nota- 
tion, while  the  yendan  refers  to  explanations  or  method  of  analysis. 
The  exact  nature  and  value  of  these  two  methods  are  not  altogether 
clear.  By  the  Chinese  "celestial  element"  method  the  roots  of  equa- 
tions were  computed  one  digit  at  a  time.  Seki  removed  this  limita- 
tion. Building  on  results  of  his  predecessors,  Seki  gives  also  rules  for 
writing  down  magic  squares  of  (2W  -t-i)^  cells.  In  the  case  of  the  more 
troublesome  even  celled  squares,  Seki  first  gives  a  rule  for  the  con- 
struction of  a  magic  square  of  4^  cells,  then  of  4(^+1)^  and  16  n^  cells. 
He  simplified  also  the  treatment  of  magic  circles.  Perhaps  the  most 
original  and  important  work  of  Seki  is  the  invention  of  determinants, 
sometime  before  1683.  Leibniz,  to  whom  the  first  idea  of  determinants 
is  usually  assigned,  made  his  discovery  in  1693  when  he  stated  that 
three  Hnear  equations  in  x  and  y  can  have  the  same  ratio  only  when 
the  determinant  arising  from  the  coefficients  vanishes.  Seki  took  n 
equations  and  gave  a  more  general  treatment.  Seki  knew  that  a 
determinant  of  the  n^^  order,  when  expanded,  has  n^  terms  and  that 
rows  and  columns  are  interchangeable.^  Usually  attributed  to  Seki 
is  the  invention  of  the  yenri  or  "circle-principle"  wnich,  it  is  claimed, 
accomplishes  somewhat  the  same  things  as  the  differential  and  in- 
tegral calculus.  Neither  the  exact  nature  nor  the  origin  of  the  yenri 
is  well  understood.  Doubt  exists  whether  Seki  was  its  discoverer. 
Takebe,  a  pupil  of  Seki,  used  the  yenri  and  may  be  the  chief  originator 

1  For  details  consult  Y.  Mikami,  "On  the  Japanese  theory  of  determinants"  in 
Isis,  "Vol.  II,  1914.  PP-  9~36- 


THE  JAPANESE  81 

of  it,  but  his  explanations  are  incomplete  and  obscu-e.  Seki,  Takebe 
and  their  co-workers  dealt  with  infinite  series,  especially  in  the  study 
of  the  circle  and  of  tt.  Probably  some  knowledge  of  European  mathe- 
matics found  its  way  into  Japan  in  the  seventeenth  century.  A 
Japanese,  under  the  name  of  Pelrus  Hartsingius,  is  known  to  have 
studied  at  Leyden  under  Van  Schooten,  but  there  is  no  clear  evidence 
that  he  returned  to  Japan.  In  1650  a  Portuguese  astronomer,  whose 
real  name  is  not  known  and  whose  adopted  name  was  Sawano  Chuan, 
translated  a  European  astronomical  work  into  Japanese.^ 

In  the  eighteenth  century  the  followers  of  Seki  were  in  control. 
Their  efforts  were  expended  upon  problem-solving,  the  mensuration 
of  the  circle  and  the  study  of  infinite  series.  Of  Kiirushima  Gita, 
who  died  in  1757,  fragmentary  manuscripts  remain,  which  show  a 
"magic  cube"  composed  of  four  4--celled  magic  squares  in  which  the 
sums  of  rows  and  columns  is  130,  and  the  sums  of  corresponding  cells 
of  the  four  squares  is  likewise  130.  This  "magic  cube"  is  evidently  a 
different  thing  from  Tanaka's  "Magic  cube."  Near  the  close  of  the 
eighteenth  century,  during  the  waning  days  of  the  Seki  school,  there 
arose  a  bitter  controversy  between  Fujita  Sadasuke,  then  the  head 
of  the  Seki  school,  and  Aida  Ammei.  Of  the  two,  Aida  was  the  younger 
and  more  gifted — an  insurgent  against  the  old  and  involved  methods 
of  exposition.  Aida  worked  on  the  approximate  solution  of  numerical 
equations.^  The  most  noted  work  of  the  time  was  done  by  a  man 
living  in  peaceful  seclusion,  Ajima  Chokuyen  of  Yedo,  who  died  in 
1798.  He  worked  on  Diophantine  analysis  and  on  a  problem  known 
in  the  West  as  "the  Malfatti  problem,"  to  inscribe  three  circles  in  a 
triangle,  each  tangent  to  the  other  two.  This  problem  appeared  in 
Japan  in  1781.  Malfatti's  publication  on  it  appeared  in  1803,  but 
the  special  case  of  the  isosceles  triangle  had  been  considered  by  Jakob 
Bernoulli  before  1744.  Ajima  treats  special  cases  of  the  problem  to 
determine  the  number  of  figures  in  the  repetend  in  circulating  decimals. 
He  improved  the  yenri  and  placed  mathematics  on  the  highest  plane 
that  it  reached  in  Japan  during  the  eighteenth  century. 

In  the  early  part  of  the  nineteenth  century  there  was  greater  in- 
filtration of  European  mathematics.  There  was  considerable  activity, 
but  no  noteworthy  names  appeared,  except  Wada  Nei  (i 787-1840) 
who  perfected  the  yenri  still  further,  developing  an  integral  calculus 
that  served  the  ordinary  purposes  of  mensuration,  and  giving  reasons 
where  his  predecessors  ordinarily  gave  only  rules.  He  worked  par- 
ticularly on  maxima  and  minima,  and  on  roulettes.  Japanese  re- 
searches of  his  day  relate  to  groups  of  ellipses  and  other  figures  which 

'  Y.  Mikami  in  Annaes  da  academia  polyt.  do  Porlo,  Vol.  VIII,  1913. 

^  Y.  Mikami,  "On  Aida  Ammei's  solution  of  an  equation"  in  Annaes  da  Academia 
Polyl.  do  Porto,  Vol.  VIII,  1913.  This  article  gives  details  on  the  solution  of  equa- 
tions in  China  and  Japan.  See  also  INIikami's  article  on  Miyai  Antai  in  the  Tohoku 
hfalh.  Jnunid.  Vol.  5,  Nos.  3,  4,  197 1. 


82  A  HISTORY  OF  MATHEMATICS 

can  be  drawn  upon  a  folding  fan.  Here  mathematics  finds  application 
to  artistic  design. 

After  the  middle  of  the  nineteenth  century  the  native  mathematics 
of  Japan  yielded  to  a  strong  influx  of  Western  mathematics.  The 
movement  in  Japan  became  a  part  of  the  great  international  advance. 
In  191 1  there  was  started  the  Tohoku  Mathematical  Journal,  under 
the  editorship  of  T.  Hayashi.  It  is  devoted  to  advanced  mathematics, 
contains  articles  in  many  of  our  leading  modern  languages  and  is  quite 
international  in  character.^ 

Looking  back  we  see  that  Japan  produced  some  able  mathemati- 
cians, but  on  account  of  her  isolation,  geographically  and  socially, 
her  scientific  output  did  not  affect  or  contribute  to  the  progress  of 
mathematics  in  the  West.  The  Babylonians,  Hindus,  Arabs,  and  to 
some  extent  even  the  Chinese  through  their  influence  on  the  Hindus, 
contributed  to  the  onward  march  of  mathematics  in  the  West.  But 
the  Japanese  stand  out  in  complete  isolation. 

^  G.  A.  Miller,  Historical  Introduction  to  Mathematical  Literature,  1916,  p.  24. 


THE  HINDUS 

After  the  time  of  the  ancient  Greeks,  the  first  people  whose  re- 
searches wielded  a  wide  influence  in  the  world  march  of  mathematics, 
belonged,  like  the  Greeks,  to  the  Aryan  race.  It  was,  however,  not  a 
European,  but  an  Asiatic  nation,  and  had  its  seat  in  far-off  India. 

Unlike  the  Greek,  Indian  society  was  fixed  into  castes.  The  only 
castes  enjoying  the  privilege  and  leisure  for  advanced  study  and 
thinking  were  the  Brahmins,  whose  prime  business  was  religion  and 
philosophy,  and  the  Kshatriyas,  who  attended  to  war  and  government. 

Of  the  development  of  Hindu  mathematics  we  know  but  Httle.  A 
few  manuscripts  bear  testimony  that  the  Indians  had  climbed  to  a 
lofty  height,  but  their  path  of  ascent  is  no  longer  traceable.  It  would 
seem  that  Greek  mathematics  grew  up  under  more  favorable  condi- 
tions than  the  Hindu,  for  in  Greece  it  attained  an  independent  exist- 
ence, and  was  studied  for  its  own  sake,  while  Hindu  mathematics 
always  remained  merely  a  servant  to  astronomy.  Furthermore,  in 
Greece  mathematics  was  a  science  of  the  people,  free  to  be  cultivated 
by  all  who  had  a  liking  for  it;  in  India,  as  in  Egypt,  it  was  in  the 
hands  chiefly  of  the  priests.  Again,  the  Indians  were  in  the  habit  of 
putting  into  verse  all  mathematical  results  they  obtained,  and  of 
clothing  them  in  obscure  and  mystic  language,  which,  though  well 
adapted  to  aid  the  memory  of  him  who  already  understood  the  subject, 
was  often  unintelligible  to  the  uninitiated.  Although  the  great  Hindu 
mathematicians  doubtless  reasoned  out  most  or  aU  of  their  discoveries, 
yet  they  were  not  in  the  habit  of  preserving  the  proofs,  so  that  the 
naked  theorems  and  processes  of  operation  are  aU  that  have  come 
down  to  our  time.  Very  different  in  these  respects  were  the  Greeks. 
Obscurity  of  language  was  generally  avoided,  and  proofs  belonged 
to  the  stock  of  knowledge  quite  as  much  as  the  theorems  themselves. 
Very  striking  was  the  difference  in  the  bent  of  mind  of  the  Hindu  and 
Greek;  for,  while  the  Greek  mind  was  pre-eminently  geometrical,  the 
Indian  was  first  of  all  arithmetical.  The  Hindu  dealt  with  number,  the 
Greek  with  form.  Numerical  symbolism,  the  science  of  numbers, 
and  algebra  attained  in  India  far  greater  perfection  than  they  had 
previously  reached  in  Greece,  On  the  other  hand,  Hindu  geometry 
was  merely  mensuration,  unaccompanied  by  demonstration.  Hindu 
trigonometry  is  meritorious,  but  rests  on  arithmetic  more  than  on 
geometry. 

An  interesting  but  difficult  task  is  the  tracing  of  the  relation  be- 
tween Hindu  and  Greek  mathematics.  It  is  well  known  that  more 
or  less  trade  was  carried  on  between  Greece  and  India  from  early 

83 


84  A  HISTORY  OF  MATHEMATICS 

times.  After  Egypt  had  become  a  Roman  province,  a  rnore  lively 
commercial  intercourse  sprang  up  between  Rome  and  India,  by  way 
of  Alexandria.  A  priori,  it  does  not  seem  improbable,  that  with  the 
traffic  of  merchandise  there  should  also  be  an  interchange  of  ideas. 
That  communications  of  thought  from  the  Hindus  to  the  Alexandrians 
actually  did  take  place,  is  evident  from  the  fact  that  certain  philo- 
sophic and  theologic  teachings  of  the  Manicheans,  Neo-Platonists, 
Gnostics,  show  unmistakable  likeness  to  Indian  tenets.  Scientific 
facts  passed  also  from  Alexandria  to  India.  This  is  shown  plainly 
by  the  Greek  origin  of  some  of  the  technical  terms  used  by  the  Hindus. 
Hindu  astronomy  was  influenced  by  Greek  astronomy.  A  part  of 
the  geometrical  knowledge  which  they  possessed  is  traceable  to  Alex- 
andria, and  to  the  writings  of  Heron  in  particular.  In  algebra  there 
was,  probably,  a  mutual  giving  and  receiving. 

There  is  evidence  also  of  an  intimate  connection  between  Indian 
and  Chinese  mathematics.  In  the  fourth  and  succeeding  centuries  of 
our  era  Indian  embassies  to  China  and  Chinese  visits  to  India  are 
recorded  by  Chinese  authorities.^  We  shall  see  that  undoubtedly 
there  was  an  influx  of  Chinese  mathematics  into  India. 

The  history  of  Hindu  mathematics  may  be  resolved  into  two 
periods:  First  the  S'ulvasutra  period  which  terminates  not  later  than 
200  A.  D.,  second,  the  astronomical  and  mathematical  period,  extending 
from  about  400  to  1200  a.  d. 

The  term  S'ulvasutra  means  "the  rules  of  the  cord";  it  is  the  name 
given  to  the  supplements  of  the  Kalpasutras  which  explain  the  con- 
struction of  sacrificial  altars.'^  The  S'ulvasutras  were  composed  some- 
time between  800  b.  c.  and  200  a.  d.  They  are  known  to  modern 
scholars  through  three  quite  modern  manuscripts.  Their  aim  is 
primarily  not  mathematical,  but  religious.  The  mathematical  parts 
relate  to  the  construction  of  squares  and  rectangles.  Strange  to  say, 
none  of  these  geometrical  constructions  occur  in  later  Hindu  works; 
later  Indian  mathematics  ignores  the  S'ulvasutras! 

The  second  period  of  Hindu  mathematics  probably  originated 
with  an  influx  from  Alexandria  of  western  astronomy.  To  the  fifth 
century  of  our  era  belongs  an  anonymous  Hindu  astronomical  work, 
called  the  Surya  Siddhanta  ("Knowledge  from  the  sun")  which  came 
to  be  regarded  a  standard  work.  In  the  sixth  century  a.  d.,  Varaha 
Mihira  wrote  his  Pahcha  Siddhantika  which  gives  a  summary  of  the 
Silrya  Siddhanta  and  four  other  astronomical  works  then  in  use;  it 
contains  matters  of  mathematical  interest. 

In  1881  there  was  found  at  Bakhshdll,  in  northwest  India,  buried 
in  the  earth,  an  anonymous  arithmetic,  supposed,  from  the  peculiar- 

1  G.  R.  Kaye,  Indian  Mathematics,  Calcutta  &  Simla,  1915,  p.  38.  We  are  draw- 
ing heavily  upon  this  book  which  embodies  the  results  of  recent  studies  of  Hindu 
mathematics. 

2  G.  R.  Kaye,  op.  cit.,  p.  .^. 


THE  HINDUS  85 

ities  of  its  verses,  to  date  from  the  third  or  fourth  century  after  Christ. 
The  document  that  was  found  is  of  birch  bark,  and  is  an  incomplete 
copy,  prepared  probably  about  the  eighth  century,  of  an  older  manu- 
script.^   It  contains  arithmetical  computation. 

The  noted  Hindu  astronomer  Aryabhata  was  bom  476  a.  d.  at 
Pataliputra,  on  the  upper  Ganges.  His  celebrity  rests  on  a  work 
entitled  Aryabhatiya,  of  which  the  third  chapter  is  devoted  to  mathe- 
matics. About  one  hundred  years  later,  mathematics  in  India  reached 
the  highest  mark.  At  that  time  flourished  Brahmagupta  (born  598). 
In  628  he  wrote  his  Brahnia-sphula-siddhanta  ("The  Revised  System 
of  Brahma"),  of  which  the  twelfth  and  eighteenth  chapters  belong 
to  mathematics. 

Probably  to  the  ninth  century  belongs  Mahavira,  a  Hindu  author 
on  elementary  mathematics,  whose  writings  have  only  recently  been 
brought  to  the  attention  of  historians.  He  is  the  author  of  the  Ganita- 
Sara-Sangraha  which  throws  light  upon  Hindu  geometry  and  arith- 
metic. The  following  centuries  produced  only  two  names  of  impor- 
tance; namely,  S'ridhara,  who  wrote  a  Ganila-sara  ("Quintessence 
of  Calculation"),  and  Padmanabha,  the  author  of  an  algebra.  The 
science  seems  to  have  made  but  little  progress  at  this  time;  for  a 
work  entitled  Siddhdnta  S'iromani  ("Diadem  of  an  Astronomical 
System"),  written  by  Bhaskara  in  1150,  stands  little  higher  than  that 
of  Brahmagupta,  written  over  500  years  earlier.  The  two  most  im- 
portant mathematical  chapters  in  this  work  are  the  Llldvatl  (  ="the 
beautiful,"  i.  e.  the  noble  science)  and  Vlja-ganita  (="  root-extrac- 
tion"), devoted  to  arithmetic  and  algebra.  From  now  on,  the  Hindus 
in  the  Brahmin  schools  seemed  to  content  themselves  with  studying 
the  masterpieces  of  their  predecessors.  Scientific  intelligence  de- 
creases continually,  and  in  more  modern  times  a  very  deficient  Arabic 
work  of  the  sixteenth  century  has  been  held  in  great  authority. 

Th©  mathematical  chapters  of  the  Brahma-slddhdnta  and  Siddhdnta 
S'iromani  were  translated  into  English  by  H.  T.  Colebrooke,  London, 
1817.  The  Surya-siddhdnta  was  translated  by  E.  Burgess,  and  anno- 
tated by  W.  D.  Whitney,  New  Haven,  Conn.,  i860,  Mahavlra's 
Ganita-Sara-Sangraha  was  published  in  191 2  in  Madras  by  M.  Ranga- 
carya. 

We  begin  with  geometry,  the  field  in  which  the  Hindus  were  least 
proficient.  The  S'ulvasiitras  indicate  that  the  Hindus,  perhaps  as 
early  as  800  b.  c,  applied  geometry  in  the  construction  of  altars. 
Kaye^  states  that  the  mathematical  rules  found  in  the  S^ulvasutras 
"  relate  to  (i)  the  construction  of  squares  and  rectangles,  (2)  the  rela- 
tion of  the  diagonal  to  the  sides,  (3)  equivalent  rectangles  and  squares. 
(4)  equivalent  circles  and  squares."    A  knowledge  of  the  Pythagorean 

^The  Bakhshali  Manuscript,  edited  by  Rudolf  Ploernly  in  the  Indian  Antiquary, 
■^^■''j  33-48  and  275-279,  Bombay,  1888. 
^  G.  R.  Kaye,  op.  cit.,  p.  4. 


86  A  HISTORY  OF  MATHEMATICS 

theorem  is  disclosed  in  such  relations  as  3^+4"  =5^,  12^+16''  =2\j^, 
15^+36^=39^.  There  is  no  evidence  that  these  expressions  were 
obtained  from  any  general  rule.  It  will  be  remembered  that  special 
cases  of  the  Pythagorean  theorem  were  known  as  early  as  1000  b.  c. 
in  China  and  as  early  as  2000  b,  c.  in  Egypt.  A  curious  expression 
for  the  relation  of  the  diagonal  to  a  square,  namely, 

V2  =1  +3+3Ti— 3.4.34  } 

is  explained  by  Kaye  as  being  "an  expression  of  a  direct  measure- 
ment" which  may  be  obtained  by  the  use  of  a  scale  of  the  kind  named 
in  one  of  the  S'ulvasutra  manuscripts,  and  based  upon  the  change 
ratios  3,  4,  34.  It  is  noteworthy  that  the  fractions  used  are  all  unit 
fractions  and  that  the  expression  yields  a  result  correct  to  five  decimal 
places.  The  S'ulvasutra  rules  yield,  by  the  aid  of  the  Pythagorean 
theorem,  constructions  for  finding  a  square  equal  to  the  sum  or  dif- 
ference of  two  squares;  they  yield  a  rectangle  equal  to  a  given  square, 
with  aV2  and  ^aVI  as  the  sides  of  the  rectangle;  they  yield  by 
geometrical  construction  a  square  equal  to  a  given  rectangle,  and 
satisfying  the  relation  ab  ={b+[a—b]/2)^-l{a-by,  corresponding 
to  Euclid  II,  5.  In  the  S'ulvasutras  the  altar  building  ritual  explains 
the  construction  of  a  square  equal  to  a  circle.  Let  a  be  the  side  of 
a  square  and  d  the  diameter  of  an  equivalent  circle,  then  the  given 
rules  may  be  expressed  thus:  ^  d^a+iayll-a)/^,  a=d-  2d/is,  a  = 
<^(i-i+8:T9-8:2ir:6+a-.2 9.6.8)-  This  third  expression  may_be  ob- 
tained from  the  first  by  the  aid  of  the  approximation  for  V2,  given 
above.  Strange  to  say,  none  of  these  geometrical  constructions  occu 
in  later  Hindu  works;  the  latter  completely  ignore  the  mathematical 
contents  of  the  S'ulvasutras.  _ 

During  the  six  centuries  from  the  time  of  Aryabhata  to  that  of 
Bhaskara,  Hindu  geometry  deals  mainly  with  mensuration.  The 
Hindu  gave  no  definitions,  no  postulates,  no  axioms,  no  logical  chain 
of  reasoning.  His  knowledge  of  mensuration  was  largely  borrowed 
from  the  Mediterranean  and  from  China,  through  imperfect  channels  of 
communication.  Aryabhata  gives  a  rule  for  the  area  of  triangle  which 
holds  only  for  the  isosceles  triangle.  Brahmagupta  distinguishes 
between  approximate  and  exact  areas,  and  gives  Heron  of  Alexan- 
dria's  famous  formula  for  the  triangular  area,  yl s{s  —  a){s-b){s  —  c). 
Heron's  formula  is  given  also  by  Mahavlra^  who  advanced  be- 
yond his  predecessors  in  giving  the  area  of  an  equilateral  triangle  as 
a'VTA-  Brahmagupta  and  MahavTra  make  a  remarkable  extension 
of  Heron's  formula  by  giving  y/ {s  —  a)(s-b){s  —  c){s—d)  as  the  area 
of  a  quadrilateral  whose  sides  are  a,  b,  c,  d,  and  whose  semiperimeter 
is  s.    That  this  formula  is  true  only  for  quadrilaterals  that  can  be  in- 

1  G.  R.  Kaye,  op.  cil.,  p.  7. 

''D.  E.  Smith,  in  Isis,  Vol.  i,  igi,^,  pp.  igg,  200. 


THE  HINDUS  87 

scribed  in  a  circle  was  recognized  by  Brahmagupta,  according  to  Can- 
tor's ^  and  Kaye's  "^  interpretation  of  Brahmagupta's  obscure  ex- 
position, but  Hindu  commentators  did  not  understand  the  limitation 
and  Bhaskara  linally  pronounced  the  formula  unsound.  Remarkable 
is  "Brahmagupta's  theorem"  on  cyclic  quadrilaterals,  x-={ad-\-b)c. 
{ac-\-bd)/(ab-\-cd)  and  y^  =  {ab-{- cd)  (ac-{-bd)/(ad-{-bc),  where  x  and 
y  are  the  diagonals  and  a,  b,  c,  d,  the  lengths  of  the  sides;  also  the  the- 
orem that,  if  a--{-b-  =  c^  and  A'-^B'^  =  C-,  then  the  quadrilateral 
("Brahmagupta's  trapezium"),  {aC,  cB,  bC,  cA)  is  cyclic  and  has  its 
diagonals  at  right  angles.  Kaye  says:  From  the  triangles  (3,  4,  5) 
and  (5,  12,  13)  a  commentator  obtains  the  quadrilateral  (39,  60,  52, 
25),  with  diagonals  63  and  56,  etc.  Brahmagupta  (says  Kaye)  also  in- 
troduces a  proof  of  Ptolemy's  theorem  and  in  doing  this  follows  Dio- 
phantus  (III,  19)  in  constructing  from  right  triangles  (a,  b,  c)  and 
(a,  /3,  y)  new  right  triangles  {ay,  by,  cy)  and  {ac,  ^c,  yc)  and 
uses  the  actual  examples  given  by  Diophantus,  namely  (39,  52,  65) 
and  (25,  60,  65).  Parallelisms  of  this  sort  show  unmistakably  that  the 
Hindus  drew  from  Greek  sources. 

_  In  the  mensuration  of  solids  remarkable  inaccuracies  occur  in 
Aryabhata.  He  gives  the  volume  of  a  pyramid  as  halj  the  product  of 
the  base  and  the  height;  the  volume  of  a  sphere  as  tt^  r^.  Aryab- 
hata gives  in  one  place  an  extremely  accurate  value  for  tt,  viz.  2>t^o 
(  =  3.1416),  but  he  himself  never  utihzed  it,  nor  did  any  other  Hindu 
mathematician  before  the  twelfth  century.  A  frequent  Indian  prac- 
tice was  to  take  7r  =  3,  or  V  10.  Bhaskara  gives  two  values, — the 
above  mentioned  'accurate,'  y||-^,  and  the  'inaccurate,'  Archimedean 
value,  ^.  A  commentator  on  Lilavatl  says  that  these  values  were 
calculated  by  beginning  with  a  regular  inscribed  hexagon,  and  apply- 
ing repeatedly  the  formula  AD=y/2  —  ^l^-AB'^,  wherein  AB  is  the 
side  of  the  given  polygon,  and  AD  that  of  one  with  double  the  number 
of  sides.  In  this  way  were  obtained  the  perimeters  of  the  inscribed 
polygons  of  12,  24,  48,  96,  192,  384  sides.  Taking  the  radius=  100,  the 
perimeter  of  the  last  one  gives  the  value  which  Aryabhata  used  for  tt. 
The  empitical  nature  of  Hindu  geometry  is  illustrated  by  Bhaskara's 
proof  of  the  Pythagorean  theorem. 
He  draws  the  right  triangle  four 
times  in  the  square  of  the  hypote- 
nuse, so  that  in  the  middle  there  re- 
mains a  square  whose  side  equals 
the  difference  between  the  two  sides 
of  the  right  triangle.  Arranging  this  square  and  the  four  triangles  in 
a  different  way,  they  are  seen,  together,  to  make  up  the  sum  of  the 
square  of  the  two  sides.     "Behold!"  says  Bhaskara,  without  adding 

1  Cantor,  op.  cit.,  Vol.  I,  3rd  Ed.,  1907,  pp.  649-653. 

*  G.  R.  Kaye,  op.  cit.,  pp.  20-32. 


88  A  HISTORY  OF  MATHEMATICS 

another  word  of  explanation.  Bretschneider  conjectures  that  the 
Pythagorean  proof  was  substantially  the  same  as  this.  Recently  it 
has  been  shown  that  this  interesting  proof  is  not  of  Hindu  origin,  but 
was  given  much  earlier  (early  in  the  Christian  era)  by  the  Chinese 
writer  Chang  Chun-Ch'ing,  in  his  commentary  upon  the  ancient  treat- 
ise, the  Chou-pei.^  In  another  place,  Bhaskara  gives  a  second  dem- 
onstration of  this  theorem  by  drawing  from  the  vertex  of  the  right 
angle  a  perpendicular  to  the  hypotenuse,  and  comparing  the  two  tri- 
angles thus  obtained  with  the  given  triangle  to  which  they  are  similar. 
This  proof  was  unknown  in  Europe  till  Wallis  rediscovered  it.  The 
only  Indian  work  that  touches  the  subject  of  the  conic  sections  is  Ma- 
havira's  book,  which  gives  an  inaccurate  treatment  of  the  ellipse.  It 
is  readily  seen  that  the  Hindus  cared  Uttle  for  geometry.  Brahma- 
gupta's  cyclic  quadrilaterals  constitute  the  only  gem  in  their  geom- 
etry. 

The  grandest  achievement  of  the  Hindus  and  the  one  which,  of  all 
mathematical  inventions,  has  contributed  most  to  the  progress  of 
intelligence,  is  the  perfecting  of  the  so-called  "Arabic  Notation." 
That  this  notation  did  not  originate  with  the  Arabs  is  now  admitted 
by  every  one.  Until  recently  the  preponderance  of  authority  favored 
the  hypothesis  that  our  numeral  system,  with  its  concept  of  local 
value  and  our  symbol  for  zero,  was  wholly  of  Hindu  origin.  Now  it 
appears  that  the  principal  of  local  value  was  used  in  the  sexagesimal 
system  found  on  Babylonian  tablets  dating  from  1600  to  2300  b.  c. 
and  that  Babylonian  records  from  the  centuries  immediately  preced- 
ing the  Christian  era  contain  a  symbol  for  zero  which,  however,  was 
not  used  in  computation.  These  sexagesimal  fractions  appear  in 
Ptolemy's  Almagest  in  130  a.  d.,  where  the  omicron  o  is  made  to  des- 
ignate blanks  in  the  sexagesimal  numbers,  but  was  not  used  as  a  reg- 
ular zero.  The  Babylonian  origin  of  the  sexagesimal  fractions  used  by 
Hindu  astronomers  is  denied  by  no  one.  The  earliest  form  of  the  In- 
dian symbol  for  zero  was  that  of  a  dot  which,  according  to  Biihler,^ 
was  "commonly  used  in  inscriptions  and  manuscripts  in  order  to 
mark  a  blank."  This  restricted  early  use  of  the  symbol  for  zero  re- 
sembles somewhat  the  still  earlier  use  made  of  it  by  the  Babylonians 
and  by  Ptolemy.  It  is  therefore  probable  that  an  imperfect  notation 
involving  the  principle  of  local  value  and  the  use  of  the  zero  was  im- 
ported into  India,  that  it  was  there  transferred  from  the  sexagesimal 
to  the  decimal  scale  and  then,  in  the  course  of  centuries,  brought  to 
final  perfection.  If  these  views  are  found  by  further  research  to  be 
correct,  then  the  name  "  Babylonic-Hindu "  notation  will  be  more 
appropriate  than  either  "Arabic"  or  "Hindu- Arabic."     It  appears 

1  Yoshio  Mikami,  "The  Pythagorean  Theorem"  in  Archiv  d.  Math.  u.  Physik, 
3.  S.,  Vol.  22,  1Q12,  pp.  1-4. 

^2  Quoted  by  D.  E.  Smith  and  L.  C.  Karpinski  in  their  Hindu-Arabic  Numerals, 
Boston  and  London,  1911,  p.  53. 


THE  HINDUS  Sg 

that  in  India  various  numeral  forms  were  used  long  before  the  prin- 
ciple of  local  value  and  the  zero  came  to  be  used.  Early  Hindu  numer- 
als have  been  classified  under  three  great  groups.  Numeral  forms  of 
one  of  these  groups  date  from  the  third  century  b.  c.^  and  are  believed 
to  be  the  forms  from  which  our  present  system  developed.  That  the 
nine  figures  were  introduced  quite  early  and  that  the  principle  of  lo- 
cal value  and  the  zero  were  incorporated  later  is  a  belief  which  re- 
ceives support  from  the  fact  that  on  the  island  of  Ceylon  a  notation 
resembling  the  Hindu,  but  without  the  zero  has  been  preserved.  We 
know  that  Buddhism  and  Indian  culture  were  transplanted  to  Ceylon 
about  the  third  century  after  Christ,  and  that  this  culture  remained 
stationary  there,  while  it  made  progress  on  the  continent.  It  seems 
highly  probable,  then,  that  the  numerals  of  Ceylon  are  the  old,  im- 
perfect numerals  of  India.  In  Ceylon,  nine  figures  were  used  for  the 
units,  nine  others  for  the  tens,  one  for  loo,  and  also  one  for  looo. 
These  20  characters  enabled  them  to  write  all  the  numbers  up  to 
9999.  Thus,  8725  would  have  been  written  with  six  signs,  represent- 
ing the  following  numbers:  8,  1000,  7,  100,  20,  5.  These  Singhalesian 
signs,  like  the  old  Hindu  numerals,  are  supposed  originally  to  have 
been  the  initial  letters  of  the  corresponding  numeral  adjectives.  There 
is  a  marked  resemblance  between  the  notation  of  Ceylon  and  the  one 
used  by  Aryabhata  in  the  first  chapter  of  his  work,  and  there  only. 
Although  the  zero  and  the  principle  of  position  were  unknown  to  the 
scholars  of  Ceylon,  they  were  probably  known  to  Aryabhata;  for,  in 
the  second  chapter,  he  gives  directions  for  extracting  the  square  and 
cube  roots,  which  seem  to  indicate  a  knowledge  of  them.  The  s}Tn- 
bol  for  zero  was  called  sunya  (the  void).  It  is  found  in  the  form  of  a 
dot  in  the  Bakhshall  arithmetic,  the  date  of  which  is  uncertain.  The 
earhest  undoubted  occurrence  of  our  zero  in  India  is  in  876  a.  d.- 
The  earliest  known  mention  of  Hindu  numerals  outside  of  India  was 
made  in  662  a.  d.  by  the  Syrian  writer,  Severus  Sebokht,  He  speaks 
of  Hindu  computations  "which  excel  the  spoken  word  and  .  .  .  are 
done  with  nine  symbols."  ^ 

There  appear  to  have  been  several  notations  in  use  in  different  parts 
of  India,  which  differed,  not  in  principle,  but  merely  in  the  forms  of 
the  signs  employed.  Of  interest  is  also  a  symbolical  system  of  position, 
in  which  the  figures  generally  were  not  expressed  by  numerical  adjec- 
tives, but  by  objects  suggesting  the  particular  numbers  in  question. 
Thus,  for  I  were  used  the  words  moon,  Brahma,  Creator,  or  form;  for 
4,  the  words  Veda,  (because  it  is  divided  into  four  parts)  or  ocean,  etc. 
The  following  example,  taken  from  the  Surya  Siddhdnta,  illustrates 
the  idea.    The  number  1,577,917,828  is  expressed  from  right  to  left  as 

^  D.  E.  Smith  and  L.  C.  KaqDinski,  op.  cit.,  p.  22. 
^  Ibid,  p.  52. 

'  M.  F.  Nau  in  Journal  Asiatique,  S.  10,  Vol.  16,  1910;  D.  E.  Smith  in  Bull.  Am. 
Math.  Soc,  Vol.  23,  1917,  p.  366. 


90  A  HISTORY  OF  MATH?:MATTCS 

follows:  Vasu  (a  class  of  8  gods) +two+eight+ mountains  (the  7  moun 
tain-chains)+form+ digits  (the  9  digits)+seven+mountains4-lunai 
days  (half  of  which  equal  15).  The  use  of  such  notations  made  it  pos- 
sible to  represent  a  number  in  several  different  ways.  This  greatly 
facilitated  the  framing  of  verses  containing  arithmetical  rules  or  sci- 
entific constants,  which  could  thus  be  more  easily  remembered. 

At  an  early  period  the  Hindus  exhibited  great  skill  in  calculating, 
even  with  large  numbers.  Thus,  they  tell  us  of  an  examination  to 
which  Buddha,  the  reformer  of  the  Indian  religion,  had  to  submit, 
when  a  youth,  in  order  to  win  the  maiden  he  loved.  In  arithmetic, 
after  having  astonished  his  examiners  by  naming  all  the  periods  of 
numbers  up  to  the  53d,  he  was  asked  whether  he  could  determine  the 
number  of  primary  atoms  which,  when  placed  one  against  the  other, 
would  form  a  line  one  mile  in  length.  Buddha  found  the  required  an- 
swer in  this  way:  7  primary  atoms  make  a  very  minute  grain  of  dust, 

7  of  these  make  a  minute  grain  of  dust,  7  of  these  a  grain  of  dust  whirled 
up  by  the  wind,  and  so  on.  Thus  he  proceeded,  step  by  step,  until  he 
finally  reached  the  length  of  a  mile.  The  multiplication  of  all  the  fac- 
tors gave  for  the  multitude  of  primary  atoms  in  a  mile  a  number  con- 
sisting of  15  digits.  This  problem  reminds  one  of  the  "  Sand-Counter" 
of  Archimedes. 

After  the  numerical  symbohsm  had  been  perfected,  figuring  was 
made  much  easier.  Many  of  the  Indian  modes  of  operation  differ 
from  ours.  The  Hindus  were  generally  inclined  to  follow  the  motion 
from  left  to  right,  as  in  writing.  Thus,  they  added  the  left-hand  col- 
umns first,  and  made  the  necessary  corrections  as  they  proceeded. 
For  instance,  they  would  have  added  254  and  663  thus:  2+6  =  8, 
5+6=11,  which  changes  8  into  9,  4+3  =  7.  Hence  the  sum  917.  In 
subtraction  they  had  two  methods.    Thus  in  821  -  348  they  would  say, 

8  from  11  =  3,4  from  11  =  7,3  fro^i  7  =  4.  Or  they  would  say,  8  from 
11  =  3^  5  from  12  =  7,  4  from  8  =  4.  In  midti plication  of  a  number  by 
another  of  only  one  digit,  say  569  by  5,  they  generally  said,  5.5  =  25, 
5.6  =  30,  which  changes  25  into  28,  5.9  =  4S>  hence  the  o  must  be  in- 
creased by  4.  The  product  is  2845.  In  the  multiplication  with  each 
other  of  many-figured  numbers,  they  first  multiplied,  in  the  manner 
just  indicated,  with  the  left-hand  digit  of  the  multiplier,  which  was 
written  above  the  multiplicand,  and  placed  the  product  above  the 
multiplier.  On  multiplying  with  the  next  digit  of  the  multiplier,  the 
product  was  not  placed  in  a  new  row,  as  with  us,  but  the  first  product 
obtained  was  corrected,  as  the  process  continued,  by  erasing,  when- 
ever necessary,  the  old  digits,  and  replacing  them  by  new  ones,  until 
finally  the  whole  product  was  obtained.  We  who  possess  the  modern 
luxuries  of  pencil  and  paper,  would  not  be  likely  to  fall  in  love  with 
this  Hindu  method.  But  the  Indians  wrote  "with  a  cane-pen  upon 
a  small  blackboard  with  a  white,  thinly  liquid  paint  which  made  marks 
that  could  be  easily  erased,  or  upon  a  white  tablet,  less  than  a  foot 


/ 

A 

/  5 

/  4 

/o 

1/ 

/    0 

THE  HINDUS  91 

square,  strewn  with  red  flour,  on  which  they  wrote  the  figures  with  a 
small  stick,  so  that  the  figures  appeared  white  on  a  red  ground."  ^ 
Since  the  digits  had  to  be  quite  large  to  be  distinctly  legible,  and 
since  the  boards  were  small,  it  was  desirable  to  have  a  method  which 
would  not  require  much  space.  Such  a  one  was  the  above  method 
of  multiplication.  Figures  could  be  easily  erased  and  replaced  by 
others  without  sacrificing  neatness.  But  the  Hindus  had  also  other 
ways  of  multiplying,  of  which  we  mention  the  following:  The  tablet 
was  divided  into  squares  like  a  chess-board.  Diagonals  were  also 
drawn,  as  seen  in  the  figure.  The  multiplication  of  12X735  =  8820  is 
exhibited  in  the  adjoining  diagram.^  According  to  Kaye,^  this  mode 
of  multiplying  was  not  of  Hindu  origin  and  was 
known  earUer  to  the  Arabs.  The  manuscripts 
extant  give  no  information  of  how  divisions  were 
executed. 

Hindu  mathematicians  of  the  twelfth  century 
test  the  correctness  of  arithmetical  computations  8       8       3      0 
by  "casting  out  nines,"  but  this  process  is  not  of  Hindu  origin; 
it  was  known  to  the  Roman  bishop  Hippolytos  in  the  third  cen- 
tury. 

In  the  Bakhshdll  arithmetic  a  knowledge  of  the  processes  of  com- 
putation is  presupposed.  In  fractions,  the  numerator  is  written  above 
the  denominator  without  a  dividing  line.  Integers  are  written  as 
fractions  with  the  denominator  i.    In  mixed  expressions  the  integral 

part  is  written  above  the  fraction.    Thus,  1  =  15.    In  place  of  our  = 

they  used  the  word  phalam,  abbreviated  into  pha.  Addition  was  in- 
dicated by  yu,  abbreviated  from  yuta.  Numbers  to  be  combined 
were  often  enclosed  in  a  rectangle.  Thus,  pha  1 2  |  f  lyu\  means  y+  j  = 
12.  An  unknown  quantity  is  siinya,  and  is  designated  thus  .  by  a 
heavy  dot.  The  word  sunya  means  "empty,"  and  is  the  word  for 
zero,  which  is  here  likewise  represented  by  a  dot.  This  double  use  of 
the  word  and  dot  rested  upon  the  idea  that  a  position  is  "empty"  if 
not  filled  out.  It  is  also  to  be  considered  "  empty  "  so  long  as  the  num- 
ber to  be  placed  there  has  not  been  ascertained.^ 

The  Bakhshall  arithmetic  contains  problems  of  which  some  are 
solved  by  reduction  to  unity  or  by  a  sort  oi  false  position.  Example: 
B  gives  twice  as  much  as  A,  C  three  times  as  much  as  B,  D  four  times  as 
much  as  C;  together  they  give  132;  how  much  did  A  give?  Take  i  for 
the  unknown  (sunya),  then  A  =1,  B  =  2,  C  =  6,  D=24,  their  sum  = 
33.    Divide  132  by  t,;^,  and  the  quotient  4  is  what  A  gave. 

The  method  of  false  position  we  have  encountered  before  among 

1  H.  Hankel,  op.  cit.,  1874,  p.  186. 

*M.  Cantor,  op.  cit..  Vol.  I,  3  Aufl.,  1907,  p.  611. 

'  G.  R.  Kaye,  op.  cit.,  p.  34. 

'■  Cantor,  I,  3  Ed.,  1907,  pp.  613-618. 


92  A  HISTORY  OF  MATHEMATICS 

the  early  Egyptians.  With  them  it  was  an  instinctive  procedure; 
with  the  Hindus  it  had  risen  to  a  conscious  method.  Bhaskara  uses 
it,  but  while  the  Bakhshall  document  preferably  assumes  i  as  the 
unknown,  Bhaskara  is  partial  to  3.  Thus,  if  a  certain  number  is 
taken  five-fold,  |  of  the  product  be  subtracted,  the  remainder  divided 
by  10,  and  |,  |  and  I  of  the  original  number  added,  then  68  is  ob- 
tained. What  is  the  number?  Choose  3,  then  you  get  15,  10,  i,  and 
i+f+i+f  =  -r-    Then  (68-=-Y-)3  =  48,  tlie  answer. 

We  shall  now  proceed  to  the  consideration  of  some  arithmetical 
problems  and  the  Indian  modes  of  solution.  A  favorite  method  was 
that  of  inversion.  With  laconic  brevity,  Aryabhata  describes  it  thus: 
"Multiphcation  becomes  division,  division  becomes  multipHcation; 
what  was  gain  becomes  loss,  what  loss,  gain;  inversion."  Quite  different 
from  this  quotation  in  style  is  the  following  problem  from  Aryabhata, 
which  illustrates  the  method:  ''Beautiful  maiden  with  beaming  eyes, 
tell  me,  as  thou  understandst  the  right  method  of  inversion,  which 
is  the  number  which  multiplied  by  3,  then  increased  by  f  of  the  prod- 
uct, divided  by  7,  diminished  by  |  of  the  quotient,  multiplied  by  it- 
self, diminished  by  52,  the  square  root  extracted,  addition  of  8,  and 
division  by  10,  gives  the  number  2?"  The  process  consists  in  begin- 
ning with  2  and  working  backwards.  Thus,  (2.10—8)^+52=196, 
\/i96=  14,  and  14.1.7.7-^3  =  28,  the  answer. 

Here  is  another  example  taken  from  Llldvati,  a  chapter  in  Bhas- 
kara's  great  work:  "The  square  root  of  half  the  number  of  bees  in  a 
swarm  has  flown  out  upon  a  jessamine-bush,  |  of  the  whole  swarm 
has  remained  behind;  one  female  bee  flies  about  a  male  that  is  buzz- 
ing within  a  lotus-flower  into  which  he  was  allured  in  the  night  by  its 
sweet  odor,  but  is  now  imprisoned  in  it.  Tell  me  the  number  of  bees." 
Answer,  72.  The  pleasing  poetic  garb  in  which  all  arithmetical  prob- 
lems are  clothed  is  due  to  the  Indian  practice  of  writing  all  school-books 
in  verse,  and  especially  to  the  fact  that  these  problems,  propounded 
as  puzzles,  were  a  favorite  social  amusement.  Says  Brahmagupta: 
"These  problems  are  proposed  simply  for  pleasure;  the  wise  man  can 
invent  a  thousand  others,  or  he  can  solve  the  problems  of  others  by 
the  rules  given  here.  As  the  sun  eclipses  the  stars  by  his  brilliancy, 
so  the  man  of  knowledge  will  eclipse  the  fame  of  others  in  assemblies 
of  the  people  if  he  proposes  algebraic  problems,  and  still  more  if  he 
solves  them." 

The  Hindus  solved  problems  in  interest,  discount,  partnership, 
alligation,  summation  of  arithmetical  and  geometric  series,  and  de- 
vised rules  for  determining  the  numbers  of  combinations  and  permu- 
tations. It  may  here  be  added  that  chess,  the  profoundest  of  all 
games,  had  its  origin  in  India.  The  invention  of  magic  squares  is 
sometimes  erroneously  attributed  to  the  Hindus.  Among  the  Chi- 
nese and  Arabs  magic  squares  appear  much  earlier.  The  first  occur- 
rence of  a  magic  square  among  the  Hindus  is  at  Dudhai,  Ihansi,  in 


THE  HINDUS 


03 


northern  India.  It  is  engraved  upon  a  stone  found  in  the  ruins  of 
a  temple  assigned  to  the  eleventh  century,  a.  d.^  After  the  time  of 
Bhaskara  magic  squares  are  mentioned  by  Hindu  writers. 

The  Hindus  made  frequent  use  of  the  "  rule  of  three."  Their  method 
of  "false  position,"  is  almost  identical  with  that  of  the  "tentative 
assumption"  of  Diophantus.  These  and  other  rules  were  applied  to 
a  large  number  of  problems. 

Passing  now  to  algebra,  we  shall  first  take  up  the  symbols  of  opera- 
tion. Addition  was  indicated  simply  by  juxtaposition  as  in  Diophan- 
tine  algebra;  subtraction,  by  placing  a  dot  over  the  subtrahend;  mul- 
tiplication, by  putting  after  the  factors,  bha,  the  abbreviation  of  the 
word  bhavita,  "the  product";  division,  by  placing  the  divisor  beneath 
the  dividend;  square-root,  by  writing  ka,  from  the  word  karana  (irra- 
tional), before  the  quantity.  The  unknown  quantity  was  called  by 
Brahmagupta  ydvatldvat  {quantum  tantum).  When  several  unknown 
quantities  occurred,  he  gave,  unlike  Diophantus,  to  each  a  distinct 
name  and  symbol.  The  first  unknown  was  designated  by  the  general 
term  "unknown  quantity."  The  rest  were  distinguished  by  names 
of  colors,  as  the  black,  blue,  yellow,  red,  or  green  unknown.  The  ini- 
tial syllable  of  each  word  constituted  the  symbol  for  the  respective 
unknown  quantity.  Thus  yd  meant  x;  kd  (from  ^d/a^a=  black)  meant 
y;  yd  kd  bha,  "x  times  y";  ka  15  ka  10,  ""v/iS— v  10." 

The  Indians  were  the  first  to  recognize  the  existence  of  absolutely 
negative  quantities.  They  brought  out  the  difference  between  posi- 
tive and  negative  quantities  by  attaching  to  the  one  the  idea  of  "pos- 
session," to  the  other  that  of  "debts."  The  conception  also  of  oppo- 
site directions  on  a  line,  as  an  interpretation  of  +  and  —  quantities, 
was  not  foreign  to  them.  They  adv^anced  beyond  Diophantus  in  ob- 
serving that  a  quadratic  has  always  two  roots.  Thus  Bhaskara  gives 
.T=5o  and  .v=— 5  for  the  roots  of  .v-  — 45.1-=  250.  "But,"  says  he, 
"  the  second  value  is  in  this  case  not  to  be  taken,  for  it  is  inadequate; 
people  do  not  approve  of  negative  roots."  Commentators  speak  of 
this  as  if  negative  roots  were  seen,  but  not  admitted. 

Another  important  generalization,  says  Hankel,  was  this,  that  since 
the  time  of  Bhaskara  the  Hindus  never  confined  their  arithmetical 
operations  to  rational  numbers.    For  instance,  Bhaskara  showed  how, 


fj         a-\-^a--b 


a—^a"  —  b 


by  the  formula  ■v/a+VT=A  "^^"  ~  +\  ~      the  square 

2  2 

root  of  the  sum  of  rational  and  irrational  numbers  could  be  found. 
The  Hindus  never  discerned  the  dividing  line  between  numbers  and 
magnitudes,  set  up  by  the  Greeks,  which,  though  the  product  of  a 
scientific  spirit,  greatly  retarded  the  progress  of  mathematics.  They 
passed  from  magnitudes  to  numbers  and  from  numbers  to  magnitudes 
without  anticipating  that  gap  which  to  a  sharply  discriminating  mind 
^  Bull.  Am.  Math.  Soc,  Vol.   24,  1917,  p.  106. 


94 


A  HISTORY  OF  MATHEMATICS 


exists  between  the  continuous  and  discontinuous.  Yet  by  doing  so 
the  Indians  greatly  aided  the  general  progress  of  mathematics.  "In- 
deed, if  one  understands  by  algebra  the  application  of  arithmetical 
operations  to  complex  magnitudes  of  all  sorts,  whether  rational  or  irra- 
tional numbers  or  space-magnitudes,  then  the  learned  Bralimins  of 
Hindostan  are  the  real  inventors  of  algebra."  ^ 

Let  us  now  examine  more  closely  the  Indian  algebra.  In  extract- 
ing the  square  and  cube  roots  they  used  the  formulas  (a+b)'  =  a^-{- 
2ab-\-b'^  and  {a-\-bY  =  a^-\- ia-b-\-  T>(^b'^'^b^ .  In  this  connection  Aryab- 
hata  speaks  of  dividing  a  number  into  periods  of  two  and  three  digits. 
From  this  we  infer  that  the  principle  of  position  and  the  zero  in  the 
numerical  notation  were  already  known  to  him.  In  figuring  with 
zeros,  a  statement  of  Bhaskara  is  interesting.  A  fraction  whose  de- 
nominator is  zero,  says  he,  admits  of  no  alteration,  though  much  be 
added  or  subtracted.  Indeed,  in  the  same  way,  no  change  takes  place 
in  the  infinite  and  immutable  Deity  when  worlds  are  destroyed  or 
created,  even  though  numerous  orders  of  beings  be  taken  up  or  brought 
forth.  Though  in  this  he  apparently  evinces  clear  mathematical  no- 
tions, yet  in  other  places  he  makes  a  complete  failure  in  figuring  with 
fractions  of  zero  denominator. 

In  the  Hindu  solutions  of  determinate  equations.  Cantor  thinks 
he  can  see  traces  of  Diophantine  methods.  Some  technical  terms  be- 
tray their  Greek  origin.  Even  if  it  be  true  that  the  Indians  borrowed 
from  the  Greeks,  they  deserve  credit  for  improving  the  solutions  of 
linear  and  quadratic  equations.  Recognizing  the  existence  of  neg- 
ative numbers,  Brahmagupta  was  able  to  unify  the  treatment  of 
the  three  forms  of  quadratic  equations  considered  by  Diophantus, 
viz.,  ax'--\-bx  =  c,  bx-{-c  =  ax,^  ax'^-\-c  =  bx,  (a,  b  and  c  being  pos- 
itive numbers),  by  bringing  the  three  under  the  one  general  case, 
px--{-qx-\-r  =  o.  To  S'ridhara  is  attributed  the  "Hindu  method" 
of  completing  the  square  which  begins  by  multiplying  both  sides 
of  the  equation  by  4/>.  Bhaskara  advances  beyond  the  Greeks 
and  even  beyond  Brahmagupta  when  he  says  that  "the  square  of 
a  positive,  as  also  of  a  negative  number,  is  positive;  that  the  square 
root  of  a  positive  number  is  twofold,  positive  and  negative.  There  is 
no  square  root  of  a  negative  number,  for  it  is  not  a  square."  Kaye 
points  out,  however,  that  the  Hindus  were  not  the  first  to  give  double 
solutions  of  quadratic  equations.^  The  Arab  Al-Khowarizmi  of  the 
ninth  century  gave  both  solutions  of  :r--i-2i  =  lox.  Of  equations  of 
higher  degrees,  the  Indians  succeeded  in  solving  only  some  special 
cases  in  which  both  sides  of  the  equation  could  be  made  perfect  powers 
by  the  addition  of  certain  terms  to  each. 

Incomparably  greater  progress  than  in  the  solution  of  determinate 
equations  was  made  by  the  Hindus  in  the  treatment  of  indeterminate 
equations.  Indeterminate  analysis  was  a  subject  to  which  the  Hindu 
1  H.  Hankel,  op.  cit.,  p.  195.  ^  G.  R.  Kaye.  op.  at.,  p.  34. 


THE  HINDUS  95 

mind  showed  a  happy  adaptation.  We  have  seen  that  this  very  sub- 
ject was  a  favorite  with  Diophantus,  and  that  his  ingenuity  was  al- 
most inexhaustible  in  devising  solutions  for  particular  cases.  But  the 
glory  of  having  invented  general  methods  in  this  most  subtle  branch 
of  mathematics  belongs  to  the  Indians.  The  Hindu  indeterminate 
analysis  differs  from  the  Greek  not  only  in  method,  but  also  in  aim. 
The  object  of  the  former  was  to  fmd  all  possible  integral  solutions. 
Greek  analysis,  on  the  other  hand,  demanded  not  necessarily  integral, 
but  simply  rational  answers.  Diophantus  was  content  with  a_single 
solution;  the  Hindus  endeavored  to  find  all  solutions  possible.  Aryab- 
hata  gives  solutions  in  integers  to  linear  equations  of  the  form  a.T=t 
by  =  c,  where  a,  b,  c  are  integers.  The  rule  employed  is  called  the  pul^ 
verizer.  For  this,  as  for  niost  other  rules,  the  Indians  give  no  proof, 
Their  solution  is  essentially  the  same  as  the  one  of  Euler.    Euler's, 

process  of  reducing  r  to  a  continued  fraction  amounts  to  the  same  as 

the  Hindu  process  of  finding  the  greatest  common  divisor  of  a  and  b 
by  division.  This  is  frequently  called  the  Diophantine  method.  Han- 
kel  protests  against  this  name,  on  the  ground  that  Diophantus  not 
only  never  knew  the  method,  but  did  not  even  aim  at  solutions  purely 
integral.^  These  equations  probably  grew  out  of  problems  in  astron- 
omy. They  were  applied,  for  instance,  to  determine  the  time  when 
a  certain  constellation  of  the  planets  would  occur  in  the  heavens. 

Passing  by  the  subject  of  linear  equations  with  more  than  two  un- 
known quantities,  we  come  to  indeterminate  quadratic  equations.  In 
the  solution  of  xy=ax-\-by-\-c,  they  applied  the  method  re-invented 
later  by  Euler,  of  decomposing  (ab-\-c)  into  the  product  of  two  integers 
m.ji  and  of  placing  x=m-\-b  and  y  =  n-\-a. 

Remarkable  is  the  Hindu  solution  of  the  quadratic  equation  cy'  = 
ax^-{-b.  With  great  keenness  of  intellect  they  recognized  in  the  special 
case  y- =  <7.v---f- 1  a  fundamental  problem  in  indeterminate  quadratics. 
They  solved  it  by  the  cyclic  method.  "It  consists,"  says  De  Morgan, 
"in  a  rule  for  finding  an  indefinite  number  of  solutions  of  y'  =  ax^-\-i 
(a  being  an  integer  which  is  not  a  square),  by  means  of  one  solution 
given  or  found,  and  of  feeling  for  one  solution  by  making  a  solution 
of  y-  =  ax--^b  give  a  solution  of  y-  =  o.v-H-6-.  It  amounts  to  the  fol- 
lowing theorem :  If  p  and  q  be  one  set  of  values  of  x  and  y  in  y-  =  ax'-\-b 
and  p'  and  q'  the  same  or  another  set,  then  qp-\-pq'  and  app'-\-qq' 
are  values  of  x  and  y  in  y'  =  ax~-\-b".  From  this  it  is  obvious  that  one 
solution  of  y-  =  <7.v--|-i  may  be  made  to  give  any  number,  and  that  if, 
taking  b  at  pleasure,  y-  =  a.v-+6-  can  be  solved  so  that  .v  and  y 
are  divisible  by  b,  then  one  preliminary  solution  of  y-  =  a.v--f-i  can  be 
be  found.  Another  mode  of  trying  for  solutions  is  a  combination  of 
the  preceding  with  the  cuttaca  (pulverizer)."  These  calculations  were 
used  in  astronomy. 

1 H.  Hankel,  op.  cii.,  p.  196. 


96  A  HISTORY  OF  MATHEMATICS 

Doubtless  this  "cyclic  method"  constitutes  the  greatest  invention 
in  the  theory  of  numbers  before  the  time  of  Lagrange.  The  perver- 
sity of  fate  has  willed  it,  that  the  equation  y~  =  ax'-\-i  should  now  be 
called  FeWs  equation;  the  first  incisive  work  on  it  is  due  to  Brahmin 
scholarship,  reinforced,  perhaps,  by  Greek  research.  It  is  a  problem 
that  has  exercised  the  highest  faculties  of  some  of  our  greatest  modern 
analysts.  By  them  the  work  of  the  Greeks  and  Hindus  was  done  over 
again ;  for,  unfortunately,  only  a  small  portion  of  the  Hindu  algebra  and 
the  Hindu  manuscripts,  which  we  now  possess,  were  known  in  the 
Occident.  Hankel  attributed  the  invention  of  the  "cyclic  method" 
entirely  to  the  Hindus,  but  later  historians,  P.  Tannery,  M.  Cantor, 
T.  Heath,  G.  R.  Kaye  favor  the  hypothesis  of  ultimate  Greek  origin. 
If  the  missing  parts  of  Diophantus  are  ever  found,  hght  will  probably 
be  thrown  upon  this  question. 

Greater  taste  than  for  geometry  was  shown  by  the  Hindus  for  trig- 
onometry. Interesting  passages  are  found  in  Varaha  Mihira's  Pahcha 
Siddhdntikd  of  the  sixth  century  a.  d.,^  which,  in  our  notation  for 
unit  radius,  gives  7r=Vio,  sin  30°=^,  sin  6o°=Vi  — j,  sin-7  = 
[sin  27)  ^/4+  [  I  — sin  (90°— 27) )  ^/4.  This  is  followed  by  a  table  of 
24  sines,  the  angles  increasing  by  intervals  of  3°45'  (the  eighth  part 
of  30°),  obviously  taken  from  Ptolemy's  table  of  chords.  However, 
instead  of  dividing  the  radius  into  60  parts  in  the  manner  of  Ptolemy, 
the  Hindu  astronomer  divides  it  into  120  parts,  which  device  enabled 
him  to  convert  Ptolemy's  table  of  chords  into  a  table  of  sines  without 
changing  the  numerical  values.  Aryabhata  took  a  still  different  value 
for  the  radius,  namely,  343S,  obtained  apparently  from  the  relation 
2X3.1416^=21,600.  The  Hindus  followed  the  Greeks  and  Babylo- 
nians in  the  practice  of  dividing  the  circle  into  quadrants,  each  quad- 
rant into  90  degrees  and  5400  minutes — thus  dividing  the  whole  circle 
into  21,600  equal  parts.  Each  quadrant  was  divided  also  into  24  equal 
parts,  so  that  each  part  embraced  225  units  of  the  whole  circumference, 
and  corresponded  to  3!  degrees.  Notable  is  the  fact  that  the  Indians 
never  reckoned,  like  the  Greeks,  with  the  whole  chord  of  double  the  arc, 
but  always  with  the  sine  (joa)  and  versed  sine.  Their  mode  of  calcula- 
ting tables  was  theoretically  very  simple.  The  sine  of  90°  was  equal  to 
the  radius,  or  3438;  the  sine  of  30°  was  evidently  half  that,  or  17 19. 


Applying  the  formula  sin^o+cos-a  =  r-,  they  obtained  sin  45°  = 

2431.    Substituting  for  cos  a  its  equal  sin  (go  — a),  and  making  a=6o°, 

they  obtained  sin  60°  =  — ^^=  2978.  With  the  sines  of  90,  60,  45,  and  30 

as  starting-points,  they  reckoned  the  sines  of  half  the  angles  by  the 

formula  ver  sin  2  a=  2  sirra,  thus  obtaining  the  sines  of  22°  30',  15°, 

^  G.  R.  Kaye,  op.  cit.,  p.  10. 


THE  HINDUS  97 

11°  15',  7°  30',  3°  45'.  They  now  figured  out  the  sines  of  the  comple- 
ments of  these  angles,  namely,  the  sines  of  86°  15',  82°  30',  78°  45', 
75°)  67°  30';  then  they  calculated  the  sines  of  half  these  angles;  then 
of  their  complements;  then,  again,  of  half  their  complements;  and  so 
on.  By  this  very  simple  process  they  got  the  sines  of  angles  at  inter- 
vals of  3°  45'.  In  this  table  they  discovered  the  unique  law  that  if 
a,  b,  c  be  three  successive  arcs  such  that  a-b  =  b—c=T,°  45',  then 

sin  a  —  sin  &  =  (sin  b  —  sin  c) .    This  formula  was  afterwards  used 

225 

whenever  a  re-calculation  of  tables  had  to  be  made.  No  Indian  trig- 
onometrical treatise  on  the  triangle  is  extant.  In  astronomy  they 
solved  plane  and  spherical  triangles.^ 

Now  that  we  have  a  fairly  complete  history  of  Chinese  mathematics, 
Kaye  has  been  able  to  point  out  parallelisms  between  Hindu  and 
Chinese  mathematics  which  indicate  that  India  is  indebted  to  China. 
The  -  Chiu-chang  Suan-shu  ("Arithmetic  in  Nine  Sections")  was  com- 
posed in  China  at  least  as  early  as  200  b.  c;  the  Chinese  writer  Chang 
Tsang  wrote  a  commentary  on  it  in  263  a.  d.  The  "Nine  Sections" 
gives  the  approximate  area  of  a  segment  of  a  circle  =  ^  {c-\-a)a,  where 
c  is  the  chord  and  a  is  the  perpendicular.  This  rule  occurs  in  the  work 
of  the  later  Hindu  author  MahavTra.  Again,  the  Chinese  problem  of 
the  bamboo  10  ft.  high,  the  upper  end  of  which  being  broken  reaches  the 
ground  three  feet  from  the  stem;  to  determine  the  height  of  the  break — 
occurs  in  all  Hindu  books  after  the  sixth  century.  The  Chinese  arith- 
metical treatise,  Sun-Tsu  Suan-ching,  of  about  the  first  century  a. 
D.  has  an  example  asking  for  a  number  which,  divided  by  3  yields  the 
remainder  2,  by  5  the  remainder  3,  and  by  7  the  remainder  2.  Exam- 
ples of  this  type  occur  in  Indian  works  of  the  seventh  and  ninth  cen- 
turies, particularly  in  Brahmagupta  and  Mahavira.  On  a  preceding 
page  we  called  attention  to  the  fact  that  Bhaskara's  dissection  proof 
of  the  Pythagorean  theorem  is  found  much  earlier  in  China.  Kaye 
gives  several  other  examples  of  Chinese  origin  that  are  found  later  in 
Hindu  books. 

Notwithstanding  the  Hindu  indebtedness  to  other  nations,  it  is 
remarkable  to  what  extent  Indian  mathematics  enters  into  the 
science  of  our  time.  Both  the  form  and  the  spirit  of  the  arithmetic 
and  algebra  of  modern  times  are  essentially  Indian.  Think  of  our 
notation  of  numbers,  brought  to  perfection  by  the  Hindus,  think  of 
the  Indian  arithmetical  operations  nearly  as  perfect  as  our  own,  think 
of  their  elegant  algebraical  methods,  and  then  judge  whether  the 
Brahmins  on  the  banks  of  the  Ganges  are  not  entitled  to  some  credit. 
Unfortunately,  some  of  the  most  brilliant  results  in  indeterminate  an- 
alysis, found  in  Hindu  works,  reached  Europe  too  late  to  exert  the  in- 

1  .\.  Arneth,  Geschichlc  d'^r  rHnm  Matkcmalik.    Stuttgart,  1852,  p.  174. 

2  G.  R.  Kaye,  op.  cil.,  pp.  38-41. 


q8  a  history  of  mathematics 

fluence  they  would  have  exerted,  had  they  come  two  or  three  centu- 
ries earher. 

At  the  beginning  of  the  twentieth  century,  mathematical  activity 
along  modern  lines  sprang  up  in  India.  In  the  year  igoy  there  was 
founded  the  Indian  Mathematical  Society;  in  1909  there  was  started 
at  Madras  the  Journal  of  the  Indian  Mathematical  Society} 

1  (Three  recent  writers  have  advanced  arguments  tending  to  disprove  the  Hindu 
origin  of  our  numerals.  We  refer  (i)  to  G.  R.  Kaye's  articles  in  Scientia,  Vol.  24, 
1918,  pp.  53-55;  in  Journal  Asiatic  Soc.  Bengal,  III,  IQ07,  pp.  475-508,  also  VII, 
lyii,  pp.  801-816:  in  Indian  Antiquary,  iqii,  pp.  50-56;  (2)  to  Carra  de  Vaux's 
article  in  Scientia,  Vol.  21,  19 17,  pp.  273-282;  (3)  to  a  Russian  book  brought  out 
by  Nikol.  Bubnow  in  1908  and  translated  into  German  in  1914  by  Jos.  Lezius. 
Kaye  claims  to  show  that  the  proofs  of  the  Hindu  origin  of  our  numerals  are 
largely  legendary,  that  the  question  has  been  clouded  by  a  confusion  between  the 
words  hindi  (Indian)  and  hindasi  (measure  geometrical),  that  the  symbols  are 
not  modified  letters  of  the  alphab&t.  We  must  hold  our  minds  in  suspense  on 
this  difficult  question  and  await  further  evidence.) 


THE  ARABS 

After  the  flight  of  Mohammed  from  Mecca  to  Medina  in  622  a.  d., 
an  obscure  people  of  Semitic  race  began  to  play  an  important  part  in 
the  drama  of  history.  Before  the  lapse  of  ten  years,  the  scattered 
tribes  of  the  Arabian  peninsula  were  fused  by  the  furnace  blast  of 
religious  enthusiasm  into  a  powerful  nation.  With  sword  in  hand  the 
united  Arabs  subdued  Syria  and  Mesopotamia.  Distant  Persia  and 
the  lands  beyond,  even  unto  India,  were  added  to  the  dominions  of 
the  Saracens.  They  concjuered  Northern  Africa,  and  nearly  the 
whole  Spanish  peninsula,  but  were  fuially  checked  from  further  prog- 
ress in  Western  Europe  by  the  firm  hand  of  Charles  Martel  (732  A.  D.). 
The  Moslem  dominion  extended  now  from  India  to  Spain;  but  a  war 
of  succession  to  the  caliphate  ensued,  and  in  755  the  Mohammedan 
empire  was  divided, — one  caliph  reigning  at  Bagdad,  the  other  at  Cor- 
dova in  Spain.  Astounding  as  was  the  grand  march  of  conquest  by 
the  Arabs,  still  more  so  was  the  ease  with  which  they  put  aside  their 
former  nomadic  life,  adopted  a  higher  civilization,  and  assumed  the 
sovereignty  over  cultivated  peoples.  Arabic  was  made  the  written 
language  throughout  the  conquered  lands.  With  the  rule  of  the  Abba- 
sides  in  the  East  began  a  new  period  in  the  history  of  learning.  The 
capital,  Bagdad,  situated  on  the  Euphrates,  lay  half-way  between 
two  old  centres  of  scientific  thought, — India  in  the  East,  and  Greece 
in  the  West.  The  Arabs  were  destined  to  be  the  custodians  of  the 
torch  of  Greek  science,  to  keep  it  ablaze  during  the  period  of  confu- 
sion and  chaos  in  the  Occident,  and  afterwards  to  pass  it  over  to  the 
Europeans.  This  remark  applies  in  part  also  to  Hindu  science.  Thus 
science  passed  from  Aryan  to  Semitic  races,  and  then  back  again  to 
the  Aryan.  Formerly  it  was  held  that  the  Arabs  added  but  little  to 
the  knowledge  of  mathematics;  recent  studies  indicate  that  they  must 
be  credited  with  novelties  once  thought  to  be  of  later  origin. 

The  Abbasides  at  Bagdad  encouraged  the  introduction  of  the 
sciences  by  inviting  able  specialists  to  their  court,  irrespective  of  na- 
tionality or  religious  belief.  Medicine  and  astronomy  were  their  fa- 
vorite sciences.  Thus  Harun-al-Rashid,  the  most  distinguished  Sara- 
cen ruler,  drew  Indian  physicians  to  Bagdad.  In  the  year  772  there 
came  to  the  court  of  Caliph  Almansur  a  Hindu  astronomer  with  as- 
tronomical tables  which  were  ordered  to  be  translated  into  Arabic. 
These  tables,  known  by  the  Arabs  as  the  Sindhind,  and  probably  taken 
from  the  Bralima-sphiita-siddhdnta  of  Brahmagupta,  stood  in  great 
authority.    They  contained  the  important  Hindu  table  of  sines. 

Doubtless  at  this  time,  and  along  with  these  astronomical  tables, 
99 


loo  A  HISTORY  OF  MATHEMATICS 

the  Hindu  numerals,  with  the  zero  and  the  principle  of  position,  were 
introduced  among  the  Saracens.  Before  the  time  of  Mohammed  the 
Arabs  had  no  numerals.  Numbers  were  written  out  in  words.  Later, 
the  numerous  computations  connected  with  the  financial  administra- 
tion over  the  conquered  lands  made  a  short  symbohsm  indispensable. 
In  some  localities,  the  numerals  of  the  more  civilized  conquered  na- 
tions were  used  for  a  time.  Thus,  in  Syria,  the  Greek  notation  was 
retained;  in  Egypt,  the  Coptic.  In  some  cases,  the  numeral  adjec- 
tives may  have  been  abbreviated  in  writing.  The  Diwani-numerals, 
found  in  an  Arabic-Persian  dictionary,  are  supposed  to  be  such  ab- 
breviations. Gradually  it  became  the  practice  to  employ  the  28  Ara- 
bic letters  of  the  alphabet  for  numerals,  in  analogy  to  the  Greek  sys- 
tem. This  notation  was  in  turn  superseded  by  the  Hindu  notation, 
which  quite  early  was  adopted  by  merchants,  and  also  by  writers  on 
arithmetic.  Its  superiority  was  generally  recognized,  except  in  as- 
tronomy, where  the  alphabetic  notation  continued  to  be  used.  Here 
the  alphabetic  notation  offered  no  great  disadvantage,  since  in  the 
sexagesimal  arithmetic,  taken  from  the  Almagest,  numbers  of  gen- 
erally only  one  or  two  places  had  to  be  written.^ 

As  regards  the  form  of  the  so-called  Arabic  numerals,  the  state- 
ment of  the  Arabic  writer  Al-Biruni  (died  1039),  who  spent  many 
years  in  India,  is  of  interest.  He  says  that  the  shape  of  the  numer- 
als, as  also  of  the  letters  in  India,  differed  in  different  localities,  and 
that  the  Arabs  selected  from  the  various  forms  the  most  suitable.  An 
Arabian  astronomer  says  there  was  among  people  much  difference  in 
the  use  of  symbols,  especially  of  those  for  5,  6,  7,  and  8.  The  symbols 
used  by  the  Arabs  can  be  traced  back  to  the  tenth  century.  We  find 
material  differences  between  those  used  by  the  Saracens  in  the  East 
and  those  used  in  the  West.  But  most  surprising  is  the  fact  that  the 
symbols  of  both  the  East  and  of  the  West  Arabs  deviate  so  extraordi- 
narily from  the  Hindu  Devanagari  numerals  (  =  divine  numerals)  of 
to-day,  and  that  they  resemble  much  more  closely  the  apices  of  the 
Roman  writer  Boethius.  This  strange  similarity  on  the  one  hand, 
and  dissimilarity  on  the  other,  is  difficult  to  explain.  The  most  plau- 
sible theory  is  the  one  of  Woepcke:  (i)  that  about  the  second  cen- 
tury after  Christ,  before  the  zero  had  been  invented,  the  Indian  nu- 
merals were  brought  to  Alexandria,  whence  they  spread  to  Rome 
and  also  to  West  Africa;  (2)  that  in  the  eighth  century,  after  the  no- 
tation in  India  had  been  already  much  modified  and  perfected  by  the 
invention  of  the  zero,  the  Arabs  at  Bagdad  got  it  from  the  Hindus; 
(3)  that  the  Arabs  of  the  West  borrowed  the  Columbus-egg,  the  zero, 
from  those  in  the  East,  but  retained  the  old  forms  of  the  nine  numer- 
als, if  for  no  other  reason,  simply  to  be  contrary  to  their  political  ene- 
mies of  the  East;  (4)  that  the  old  forms  were  remembered  by  the  West- 
Arabs  to  be  of  Indian  origin,  and  were  hence  called  Gubar-numerals 
^  H.  Hankel,  op.  cit.,  p.  255. 


THE  ARABS  loi 

(  =  dust-numerals,  in  memory  of  the  Brahmin  practice  of  reckoning 
on  tablets  strewn  with  dust  or  sand;  (5)  that,  since  the  eighth  cen- 
tury, the  numerals  in  India  underwent  further  changes,  and  assumed 
the  greatly  modified  forms  of  the  modern  Devanagari-numerals.^  This 
is  rather  a  bold  theory,  but,  whether  true  or  not,  it  explains  better 
than  any  other  yet  propounded,  the  relations  between  the  apices,  the 
Gubar,  the  East-Arabic,  and  Devanagari  numerals. 

It  has  been  mentioned  that  in  772  the  Indian  Siddhdnla  was  brought 
to  Bagdad  and  there  translated  into  Arabic.  There  is  no  evidence  that 
any  intercourse  existed  between  Arabic  and  Indian  astronomers  either 
before  or  after  this  time,  excepting  the  travels  of  Al-Binmi.  But 
w^e  should  be  very  slow  to  deny  the  probability  that  more  extended 
communications  actually  did  take  place. 

Better  informed  are  we  regarding  the  way  in  which  Greek  science, 
in  successive  waves,  dashed  upon  and  penetrated  Arabic  soil.  In 
Syria  the  sciences,  especially  philosophy  and  medicine,  were  culti- 
vated by  Greek  Christians.  Celebrated  were  the  schools  at  Antioch 
and  Emesa,  and,  first  of  all,  the  flourishing  Nestorian  school  at  Edessa. 
From  Syria,  Greek  physicians  and  scholars  were  called  to  Bagdad. 
Translations  of  works  from  the  Greek  began  to  be  made.  A  large 
number  of  Greek  manuscripts  were  secured  by  Caliph  Al-Mamun  (813- 
833)  from  the  emperor  in  Constantinople  and  were  turned  over  to 
Syria.  The  successors  oi  Al-Mamun  continued  the  work  so  auspi- 
ciously begun,  until,  at  the  beginning  of  the  tenth  century,  the  more 
important  philosophic,  medical,  mathematical,  and  astronomical 
works  of  the  Greeks  could  all  be  read  in  the  Arabic  tongue.  The  trans- 
lations of  mathematical  works  must  have  been  very  deficient  at  first, 
as  it  was  evidently  difficult  to  secure  translators  who  were  masters  of 
both  the  Greek  and  Arabic  and  at  the  same  time  proficient  in  mathe- 
matics. The  translations  had  to  be  revised  again  and  again  before 
they  were  satisfactory.  The  first  Greek  authors  made  to  speak  in 
Arabic  were  Euclid  and  Ptolemy.  This  was  accomplished  during  the 
reign  of  the  famous  Harun-al-Rashid.  A  revised  translation  of  Eu- 
clid's Elements  was  ordered  by  Al-Mamun.  As  this  revision  still  con- 
tained numerous  errors,  a  new  translation  w-as  made,  either  by  the 
learned  Hunain  ibn  Ishak,  or  by  his  son,  Ishak  ibn  Hunain.  The 
word  "ibn "  means  " son."  To  the  thirteen  books  of  the  Elements  were 
added  the  fourteenth,  written  by  H}qosicles,  and  the  fifteenth  attrib- 
uted by  some  to  Damascius.  But  it  remained  for  Tabit  ibn  Korra  to 
bring  forth  an  Arabic  Euclid  satisfying  every  need.  Still  greater  dif- 
ficulty was  experienced  in  securing  an  intelligent  translation  of  the 
Almagest.  Among  other  important  translations  into  Arabic  were  the 
works  of  Apollonius,  iVrchimedes,  Heron,  and  Diophantus.  Thus  we 
see  that  in  the  course  of  one  century  the  Arabs  gained  access  to  the 
vast  treasures  of  Greek  science. 

1  M.  Cantor,  op.  cit.,  Vol.  I,  1907,  p.  711. 


J02  A  HISTORY  OF  MATHEMATICS 

In  astronomy  great  activity  in  original  research  existed  as  early  as 
the  ninth  century.  The  religious  observances  demanded  by  Moham- 
medanism presented  to  astronomers  several  practical  problems.  The 
Moslem  dominions  being  of  such  enormous  extent,  it  remained  in 
some  localities  for  the  astronomer  to  determine  which  way  the  "'Be- 
liever" must  turn  during  prayer  that  he  may  be  facing  Mecca.  The 
prayers  and  ablutions  had  to  take  place  at  definite  hours  during  the 
day  and  night.  This  led  to  more  accurate  determinations  of  time.  To 
fix  the  exact  date  for  the  Mohammedan  feasts  it  became  neces- 
sary to  observe  more  closely  the  motions  of  the  moon.  In  addition  to 
all  this,  the  old  Oriental  superstition  that  extraordinary  occurrences 
in  the  heavens  in  some  mysterious  way  affect  the  progress  of  human 
affairs  added  increased  interest  to  the  prediction  of  eclipses.^ 

For  these  reasons  considerable  progress  was  made.  Astronomical 
tables  and  instruments  were  perfected,  observatories  erected,  and  a 
connected  series  of  observations  instituted.  This  intense  love  for  as- 
tronomy and  astrology  continued  during  the  whole  Arabic  scientific 
period.  As  in  India,  so  here,  we  hardly  ever  find  a  man  exclusively 
devoted  to  pure  mathematics.  Most  of  the  so-called  mathematicians 
were  first  of  all  astronomers. 

The  first  notable  author  of  mathematical  books  was  Mohammed 
ibn  Musa  Ai-Khowarizmi,  who  lived  during  the  reign  of  Caliph  Al- 
Mamun  (S13-S33).  Our  chief  source  of  information  about  Al-Khow- 
rizmi  is  the  book  of  chronicles,  entitled  Kitab  Al-Fihrist,  written  by 
Al-Nadim,  about  9S7  a.  d.,  and  containing  biographies  of  learned 
men.  Al-Khowarizmi  was  engaged  by  the  caliph  in  making  extracts 
from  the  Sindhind,  in  revising  the  tablets  of  Ptolemy,  in  taking  ob- 
servations at  Bagdad  and  Damascus,  and  in  measuring  a  degree  of 
the  earth's  meridian.  Important  to  us  is  his  work  on  algebra  and 
arithmetic.  The  portion  on  arithmetic  is  not  extant  in  the  original, 
and  it  was  not  till  1S57  that  a  Latin  translation  of  it  was  found.  It 
begins  thus:  "Spoken  has  Algoritmi.  Let  us  give  deserved  praise  to 
God,  our  leader  and  defender."  Here  the  name  of  the  author,  Al- 
Khowarizmi  has  passed  into  Algorilmi,  from  which  come  our  modern 
word  algorithm,  signifying  the  art  of  computing  in  any  particular 
way,  and  the  obsolete  form  augrim,  used  by  Chaucer.^  The  arith- 
metic of  Khowarizmi,  being  based  on  the  principle  of  position  and 
the  Hindu  method  of  calculation,  "excels,"  says  an  Arabic  writer, 
"all  others  in  brevity  and  easiness,  and  exhibits  the  Hindu  intellect 
and  sagacity  in  the  grandest  inventions."  This  book  was  followed 
by  a  large  number  of  arithmetics  by  later  authors,  which  differed 
from  the  earlier  ones  chiefly  in  the  greater  variety  of  methods.  Ara- 
bian arithmetics  generally  contained  the  four  operations  with  inte- 

1  H.  Hankel,  op.  cil.,  pp.  226-228. 

2  See  L.  C.  Karpinski,  "Augrimstones"  in  Modern  Language  Notes,  Vol.  27, 
:oi2,  pp.  206-200. 


THE  ARABS  103 

gers  and  fractions,  modelled  after  the  Indian  processes.  They  ex- 
plained the  operation  of  casting  out  the  g's,  also  the  regula  falsa  and 
the  regula  duorum  falsorum,  sometimes  called  the  rules  of  "false  po- 
sition" and  of  "double  position"  or  "double  false  position,"  by  which 
algebraical  examples  could  be  solved  without  algebra.  The  regula 
falsa  or  falsa  positio  was  the  assigning  of  an  assumed  value  to  the 
unknown  quantity,  which  value,  if  wrong,  was  corrected  by  some 
process  hke  the  "rule  of  three."  It  was  known  to  the  Hindus  and  to 
the  Egyptian  Ahmes.  Diophantus  used  a  method  almost  identical 
with  this.  The  regula  duorum  falsorum  M'as  as  follows:^  To  solve  an 
equation /(.t)  =  V,  assume,  for  the  moment,  two  values  for  x;  namely, 
x  =  a  and  x  =  b.    Then  form  f{a)  =  A  and  f{b)  =  B,  and  determine  the 

errors  V—A  =  Ea  and  V—B  =  Eb,  then  the  required  x  =  —^ — =--*  is 

generally  a  close  approximation,  but  is  absolutely  accurate  whenever 
/(.v)  is  a  linear  function  of  x. 

We  now  return  to  Khowarizmi,  and  consider  the  other  part  of  his 
work, — the  algebra.  This  is  the  first  book  known  to  contain  this  word 
itself  as  title.  Really  the  title  consists  of  two  words,  al-jebr  w'almu- 
qabala,  the  nearest  EngUsh  translation  of  which  is  "restoration  and 
reduction."  By  "restoration"  was  meant  the  transposing  of  negative 
terms  to  the  other  side  of  the  equation;  by  "reduction,"  the  uniting  of 
similar  terms.  Thus,  x'^  —  2x=sx-\-6  passes  by  al-jebr  into  .v-=5.v+ 
2.r+6;  and  this,  by  almuqabala,  into  x'='jx-{-6.  The  work  on  alge- 
bra, Hke  the  arithmetic,  by  the  same  author,  contains  little  that  is 
original.  It  explains  the  elementary  operations  and  the  solutions  of 
linear  and  quadratic  equations.  From  whom  did  the  author  borrow 
his  knowledge  of  algebra?  That  it  came  entirely  from  Indian  sources 
is  impossible,  for  the  Hindus  had  no  rules  like  the  "restoration"  and 
"reduction."  They  were,  for  instance,  never  in  the  habit  of  making 
all  terms  in  an  equation  positive,  as  is  done  by  the  process  of  "restora- 
tion." Diophantus  gives  two  rules  which  resemble  somewhat  those 
of  our  Arabic  author,  but  the  probability  that  the  Arab  got  all  his  al- 
gebra from  Diophantus  is  lessened  by  the  considerations  that  he  rec- 
ognized both  roots  of  a  quadratic,  while  Diophantus  noticed  only  one; 
and  that  the  Greek  algebraist,  unlike  the  Arab,  habitually  rejected 
irrational  solutions.  It  would  seem,  therefore,  that  the  algebra  of 
Al-Khowarizmi  was  neither  purely  Indian  nor  purely  Greek.  Al- 
Khowarizimi's  fame  among  the  Arabs  was  great.  He  gave  the  ex- 
amples .r-+io.v  =  39,  .x-+2i  =  IDA-,  3,vH-4  =  ,r''  which  are  used  by  later 
authors,  for  instance,  by  the  poet  and  mathematician  Omar  Khayyam. 
"The  equation  x-+ 10^  =  39  runs  like  a  thread  of  gold  through  the 
algebras  of  several  centuries"  (L.  C.  Karpinski).  It  appears  in  the 
algebra  of  Abu  Kamil  who  drew  extensively  upon  the  work  of  Al- 

1 H.  Hankel,  op.  cU.,  p.  259. 


I04  A  HISTORY  OF  MATHEMATICS 

Khowarizmi.  Abu  Kamil,  in  turn,  was  the  source  largely  drawn  upon 
by  the  Italian,  Leonardo  of  Pisa,  in  his  book  of  1202. 

The  algebra  of  Al-Khowarizmi  contains  also  a  few  meagre  frag- 
ments on  geometry.  He  gives  the  theorem  of  the  right  triangle,  but 
proves  it  after  Hindu  fashion  and  only  for  the  simplest  case,  when  the 
right  triangle  is  isosceles.  He  then  calculates  the  areas  of  the  tri- 
angle, parallelogram,  and  circle.  For  tt  he  uses  the  value  3!,  and  also 
the  two  Indian,  7r=  V 10  and  7r  =  |f  gf  |.  Strange  to  say,  the  last  value 
was  afterwards  forgotten  by  the  Arabs,  and  replaced  by  others  less 
iccurate.  Al-Khowarizmi  prepared  astronomical  tables,  which,  about 
1000  A.  D.,  were  revised  by  Maslama  al-Majrlii  and  are  of  importance 
as  containing  not  only  the  sine  function,  but  also  the  tangent  function.^ 
The  former  is  evidently  of  Hindu  origin,  the  latter  may  be  an  addi- 
aon  made  by  Maslama  and  was  formerly  attributed  to  Abu'l  Wefa. 

Next  to  be  noticed  are  the  three  sons  of  Musa  Sakir,  who  lived  in 
Bagdad  at  the  court  of  the  Caliph  Al-Mamun.  They  wrote  several 
works,  of  which  we  mention  a  geometry  containing  the  well-known 
formula  for  the  area  of  a  triangle  expressed  in  terms  of  its  sides.  We 
are  told  that  one  of  the  sons  travelled  to  Greece,  probably  to  collect 
astronomical  and  mathematical  manuscripts,  and  that  on  his  way  back 
he  made  acquaintance  with  Tabit  ibn  Korra.  Recognizing  in  him  a 
talented  and  learned  astronomer,  Mohammed  procured  for  him  a  place 
among  the  astronomers  at  the  court  in  Bagdad.  Tabit  ibn  Korra 
(836-901)  was  born  at  Harran  in  Mesopotamia.  He  was  proficient 
not  only  in  astronomy  and  mathematics,  but  also  in  the  Greek,  Arabic, 
and  Syrian  languages.  His  translations  of  Apollonius,  Archimedes, 
Euclid,  Ptolemy,  Theodosius,  rank  among  the  best.  His  dissertation 
on  amicable  numbers  (of  which  each  is  the  sum  of  the  factors  of  the 
other)  is  the  first  known  specimen  of  original  work  in  mathematics  on 
Arabic  soil.  It  shows  that  he  was  familiar  with  the  Pythagorean  the- 
ory of  numbers.  Tabit  invented  the  following  rule  for  finding  amicable 
numbers,  which  is  related  to  Euclid's  rule  for  perfect  numbers:  If 
/>  =  3.2"  — I,  9  =  3.2""'  — I,  r  =  g.2^"—^—i  (n  being  a  whole  number) 
are  three  primes,  then  a=  2"pq,  b=2^r  are  a  pair  of  amicable  numbers. 
Thus,  if  w=  2,  then/'=ii,  ?  =  5,  ^=71,  and  a=22o,  6=284.  Tabit 
also  trisected  an  angle. 

Tabit  ibn  Korra  is  the  earliest  writer  outside  of  China  to  discuss 
magic  squares.  Other  Arabic  tracts  on  this  subject  are  due  to  Ibn 
Al-Haitam  and  later  writers.^ 

Foremost  among  the  astronomers  of  the  ninth  century  ranked  AI- 

1  See  H.  Suter,  "Die  astronomischen  Tafeln  des  Muhammed  ibn  MiJsa  Al- 
Khwarizmi  in  der  Bearbeitung  des  Maslama  ibn  xAhmed  Al-Madjriti  und  der  Latein. 
Uebersetzung  des  Athelhard  von  Bath,"  in  Memoir's  de  I'AcadSmie  R.  des  Sciences 
et  des  Lettres  de  Danemark,  Copenhague,  yme  g.,  Section  des  Lettres,  t.  Ill,  no.  i, 
1914. 

2  See  H.  Suter,  Die  Mathematiker  u.  Astrorwmen  der  Arahcr  u.  Hire  Werke,  1900, 
PP-  36,  93.  136.  139.  140,  146,  2tS. 


THE  ARABS  105 

Battani,  called  Alhategnius  by  the  Latins.  Bat  tan  in  Syria  was  his 
birthplace.  His  observations  were  celebrated  for  great  precission. 
His  work,  De  scientia  stellarum,  was  translated  into  Latin  by  Plato 
Tiburtinus,  in  the  twelfth  century.  Out  of  this  translation  sprang  the 
word  "sinus,"  as  the  name  of  a  trigonometric  function.  The  Arabic 
w^ord  for  "sine,"  jiba  was  derived  from  the  Sanskrit  jzm,  and  resem- 
bled the  Arabic  word  jaib,  meaning  an  indentation  or  gulf.  Hence 
the  Latin  ''sinus."  ^  Al-Battani  was  a  close  student  of  Ptolemy,  but 
did  not  follow  him  altogether.  He  took  an  important  step  for  the 
better,  when  he  introduced  the  Indian  "sine"  or  Jmtf  the  chord,  in 
place  of  the  whole  chord  of  Ptolemy.  He  was  the  first  to  prepare  a 
table  of  cotangents.  He  dealt  with  horizontal  and  also  vertical  sun 
dials,  and  accordingly  considered  a  horizontal  shadow  (umbra  extensa 
in  Latin  translation)  and  vertical  shadow  {umbra  versa).  These  de- 
noted, respectively,  the  "cotangent"  and  "tangent";  the  former 
came  to  be  called  umbra  recta  by  Latin  writers.  Al-Battani  probably 
knew  the  law  of  sines;  that  this  law  was  known  to  Al-Biruni  is  certain. 
Another  improvement  on  Greek  trigonometry  made  by  the  Arabs 
points  likewise  to  Indian  influences.  Propositions  and  operations 
which  were  treated  by  the  Greeks  geometrically  are  expressed  by  the 
Arabs  algebraically.    Thus,  Al-Battani  at  once  gets  from  an  equation 

-n  =  D,  the  value  of  d  by  means  of  sin^=  , — r-Rb , — a  process 

cos^  si-\-D^ 

unknown  to  the  ancients.  He  knows  all  the  formulas  for  spherical 
triangles  given  in  the  Almagest,  but  goes  further,  and  adds  an  impor- 
tant one  of  his  own  for  obHque-angled  triangles;  namely,  cos  fl  =  cos6. 
cos  c-\r  sin  b  sin  c  cos  A . 

At  the  beginning  or  the  tenth  century  political  troubles  arose  in  the 
East,  and  as  a  result  the  house  of  the  Abbasides  lost  power.  One  prov- 
ince after  another  was  taken,  till,  in  945,  all  possessions  were  wrested 
from  them.  Fortunately,  the  new  rulers  at  Bagdad,  the  Persian  Buy- 
ides,  were  as  much  interested  in  astronomy  as  their  predecessors.  The 
progress  of  the  sciences  was  not  only  unchecked,  but  the  conditions 
for  it  became  even  more  favorable.  The  Emir  Adud-ed-daula  (978- 
9S3)  gloried  in  having  studied  astronomy  himself.  His  son  Saraf-ed- 
daula  erected  an  observatory  in  the  garden  of  his  palace,  and  called 
thither  a  whole  group  of  scholars.-  Among  them  were  Abu'l-Wefa, 
Ai-KtiJii,  Al-Sagani. 

Abu'I  Wefa  (940-998)  was  born  at  Buzshan  in  Chorassan,  a  region 
among  the  Persian  mountains,  which  has  brought  forth  many  Arabic 
astronomers.  He  made  the  brilliant  discovery  of  the  variation  of  the 
moon,  an  inequality  usually  supposed  to  have  been  first  discovered  by 
Tycho  Brahe.    Abu'1-Wefa  translated  Diophantus.    He  is  one  of  the 

1  M.  Cantor,  op.  cit.,  Vol.  I,  3  Aufl.,  1907,  p.  737. 
*H.  Hankel,  op.  cit.,  p.  242. 


io6  A  HISTORY  OF  MATHEMATICS 

last  Arabic  translators  and  commentators  of  Greek  authors.  The  fact 
that  he  esteemed  the  algebra  of  Mohammed  ibn  Musa  Al-Khowarizimi 
worthy  of  his  commentary  indicates  that  thus  far  algebra  had  made 
little  or  no  progress  on  Arabic  soil.  Abu'1-Wefa  invented  a  method  for 
computing  tables  of  sines  which  gives  the  sine  of  half  a  degree  correct 
to  nine  decimal  places.  He  used  the  tangent  and  calculated  a  table  of 
tangents.  In  considering  the  shadow-triangle  of  sun-dials  he  intro- 
duced also  the  secant  and  cosecant.  Unfortunately,  these  new  trigo- 
nometric functions  and  the  discovery  of  the  moon's  variation  ex- 
cited apparently  no  notice  among  his  contemporaries  and  followers. 
A  treatise  by  Abu'1-Wefa  on  "geometric  constructions"  indicates  that 
efforts  were  being  made  at  that  time  to  improve  draughting.  It  con- 
tains a  neat  construction  of  the  corners  of  the  regular  polyedrons  on 
the  circumscribed  sphere.  Here,  for  the  first  time,  appears  the  con- 
dition which  afterwards  became  very  famous  in  the  Occident,  that 
the  construction  be  effected  with  a  single  opening  of  the  compasses. 

Al-Kuhi,  the  second  astronomer  at  the  observatory  of  the  emir  at 
Bagdad,  was  a  close  student  of  Archimedes  and  Apollonius.  He  solved 
the  problem,  to  construct  a  segment  of  a  sphere  equal  in  volume  to 
a  given  segment  and  having  a  curved  surface  equal  in  area  to  that  of 
another  given  segment.  He,  Al-Sagani,  and  Al-Biruni  made  a  study 
of  the  trisection  of  angles.  Abu'l  Jud,  an  able  geometer,  solved  the 
problem  by  the  intersection  of  a  parabola  with  an  equilateral  hyper- 
bola. 

The  Arabs  had  already  discovered  the  theorem  that  the  sum  of  two 
cubes  can  never  be  a  cube.  This  is  a  special  case  of  the  "last  theorem 
of  Fermat."  Abu  Mohammed  Al-Khojandi  of  Chorassan  thought  he 
had  proved  this.  His  proof,  now  lost,  is  said  to  have  been  defective. 
Several  centuries  later  Beha-Eddin  declared  the  impossibility  of 
x^-\-y^  =  z^.  Creditable  work  in  theory  of  numbers  and  algebra  was 
done  by  Al-Karkhi  of  Bagdad,  who  lived  at  the  beginning  of  the  elev- 
enth century.  His  treatise  on  algebra  is  the  greatest  algebraic  work 
of  the  Arabs.  In  it  he  appears  as  a  disciple  of  Diophantus.  He  was 
the  first  to  operate  with  higher  roots  and  to  solve  equations  of  the 
form  x~"-\-ax^  =  h.  For  the  solution  of  quadratic  equations  he  gives 
both  arithmetical  and  geometrical  proofs.  He  was  the  first  Arabic 
author  to  give  and  prove  the  theorems  on  the  summation  of  the  se- 
ries:— 

i^+22+3^+...+  ;r=  (1+2+...+  //)^, 

r'+2^+3'+-+«'^=  (i+2+...+  «)^ 

Al-Karkhi  also  busied  himself  with  indeterminate  analysis.  He 
showed  skill  in  handling  the  methods  of  Diophantus,  but  added  no- 
thing whatever  to  the  stock  of  knowledge  already  on  hand.  Rather 
surprising  is  the  fact  that  Al-Karkhi's  algebra  shows  no  traces  what- 


THE  ARABS  107 

ever  of  Hindu  indeterminate  analysis.  But  most  astonishing  it  is, 
that  an  arithmetic  by  the  same  author  completely  excludes  the  Hindu 
numerals.  It  is  constructed  wholly  after  Greek  pattern.  Abu'1-Wefa, 
also,  in  the  second  half  of  the  tenth  century,  wrote  an  arithmetic  in 
which  Hindu  numerals  find  no  place.  This  practice  is  the  very  oppo- 
site to  that  of  other  Arabian  authors.  The  question,  why  the  Hindu 
numerals  were  ignored  by  so  eminent  authors,  is  certainly  a  puzzle. 
Cantor  suggests  that  at  one  time  there  may  have  been  rival  schools, 
of  which  one  followed  almost  exclusively  Greek  mathematics,  the 
other  Indian. 

The  Arabs  were  familiar  with  geometric  solutions  of  quadratic  equa- 
tions. Attempts  were  now  made  to  solve  cubic  equations  geometri- 
cally. They  were  led  to  such  solutions  by  the  study  of  questions  like 
the  Archimedean  problem,  demanding  the  section  of  a  sphere  by  a 
plane  so  that  the  two  segments  shall  be  in  a  prescribed  ratio.  The 
first  to  state  this  problem  in  form  of  a  cubic  equation  was  Al-Mahani 
of  Bagdad,  while  Abu  Ja'far  Alchazin  was  the  first  Arab  to  solve  the 
equation  by  conic  sections.  Solutions  were  given  also  by  Al-Kuhi, 
Al-Hasan  ibn  Al-Haitam,  and  others.  Another  difficult  problem,  to 
determine  the  side  of  a  regular  heptagon,  required  the  construction  of 
the  side  from  the  equation  x^—x'^  —  2x-\-i  =  o.  It  was  attempted  by 
many  and  at  last  solved  by  Abu'l  Jud. 

The  one  who  did  most  to  elevate  to  a  method  the  solution  of  alge- 
braic equations  by  intersecting  conies,  was  the  poet  Omar  'Kha.yyaiQ. 
of  Chorassan  (about  1045-1123).  He  divides  cubics  into  two  classes, 
the  trinomial  and  quadrinomial,  and  each  class  into  families  and  spe- 
cies. Each  species  is  treated  separately  but  according  to  a  general 
plan.  He  beUeved  that  cubics  could  not  be  soh^ed  by  calculation,  nor 
bi-quadratics  by  geometry.  He  rejected  negative  roots  and  often 
failed  to  discover  all  the  positive  ones.  Attempts  at  bi-quadratic 
equations  were  made  by  Abu'1-Wef a,  ^  who  solved  geometrically  x^  =  a 
and  x^-{-ax^  =  b. 

The  solution  of  cubic  equations  by  intersecting  conies  was  the  great- 
est achievement  of  the  Arabs  in  algebra.  The  foundation  to  this  work 
had  been  laid  by  the  Greeks,  for  it  was  IMenaechmus  who  first  con- 
structed the  roots  of  x^  —  a  =  o  or  x^— 20^  =  0.  It  was  not  his  aim  to 
find  the  number  corresponding  to  x,  but  simply  to  determine  the  side 
X  of  a  cube  double  another  cube  of  side  a.  The  Arabs,  on  the  other 
hand,  had  another  object  in  view:  to  find  the  roots  of  given  numerical 
equations.  In  the  Occident,  the  Arabic  solutions  of  cubics  remained 
unknown  until  quite  recently.  Descartes  and  Thomas  Baker  invented 
these  constructions  anew.  The  works  of  Al-Khayyam,  Al-Karkhi, 
Abu'l  Jud,  show  how  the  Arabs  departed  further  and  further  from 

'  L.  Matthiessen,  Grundziige  der  Antiken  und  Modernen  Algebra  der  Litieralen 
Glricliuugen,  Leipzif,',  1878,  p.  923.  Ludwig  Matthiessen  (1830-1906)  was  professor 
of  physics  at  Rostock. 


io8  A  HISTORY  OF  MATHEMATICS 

the  Indian  methods,  and  placed  themselves  more  immediately  under 
Greek  influences. 

With  Al-Karkhi  and  Omar  Khayyam,  mathematics  among  the 
Arabs  of  the  East  reached  flood-mark,  and  now  it  begins  to  ebb.  Be- 
tween iioo  and  130G  a.  d.  come  the  crusades  with  war  and  bloodshed, 
during  which  European  Christians  profited  much  by  their  contact  with 
Arabian  culture,  then  far  superior  to  their  own.  The  crusaders  were 
not  the  only  adversaries  of  the  Arabs.  During  the  first  half  of  the 
thirteenth  century,  they  had  to  encounter  the  wild  Mongolian  hordes, 
and,  in  1256,  were  conquered  by  them  under  the  leadership  of  Hulagu. 
The  caliphate  at  Bagdad  now  ceased  to  exist.  At  the  close  of  the  four- 
teenth century  still  another  empire  was  formed  by  Timur  or  Tamer- 
lane, the  Tartar.  During  such  sweeping  turmoil,  it  is  not  surprising 
that  science  dechned.  Indeed,  it  is  a  marvel  that  it  existed  at  aU. 
During  the  supremacy  of  Hulagu,  lived  Nasir-Eddin  (1201-1274), 
a  man  of  broad  culture  and  an  able  astronomer.  He  persuaded  Hu- 
lagu to  build  him  and  his  associates  a  large  observatory  at  Maraga. 
Treatises  on  algebra,  geometry,  arithmetic,  and  a  translation  of  Eu- 
clid's Elements,  were  prepared  by  him.  He  for  the  first  time  elabo- 
rated trigonometry  independently  of  astronomy  and  to  such  great 
perfection  that,  had  his  work  been  known,  Europeans  of  the  fifteenth 
century  might  have  spared  their  labors.^  He  tried  his  skill  at  a  proof 
of  the  parallel-postulate.  His  proof  assumes  that  if  AB  is  perpendic- 
ular to  CD  at  C,  and  if  another  straight  line  EDF  makes  an  angle 
EDC  acute,  then  the  perpendiculars  to  AB,  comprehended  between 
AB  and  EF,  and  drawn  on  the  side  of  CD  toward  E,  are  shorter  and 
shorter,  the  further  they  are  from  CD.  His  proof,  in  Latin  translation, 
was  pubhshed  by  Wallis  in  1651.^  Even  at  the  court  of  Tamerlane  in 
D  Samarkand,  the  sciences  were  by 

~~      no  means  neglected.     A  group  of 
astronomers   was   drawn  to   this 

—  B  court.    Uleg  Beg  (1393-1449),  a 
C  grandson  of  Tamerlane,  was  him- 

self an  astronomer.  Most  prominent  at  this  time  was  Al-Kashi,  the 
author  of  an  arithmetic.  Thus,  during  intervals  of  peace,  science 
continued  to  be  cultivated  in  the  East  for  several  centuries.  The 
last  Oriental  writer  was  Beha-Eddin  (1547-1622).  His  Essence  of 
Arithmetic  stands  on  about  the  same  level  as  the  work  of  Mohammed 
ibn  Musa  Khowarizmi,  written  nearly  800  years  before. 

"Wonderful  is  the  expansive  power  of  Oriental  peoples,  with  which 
upon  the  wings  of  the  wind  they  conquer  half  the  world,  but  more 
wonderful  the  energy  with  which,  in  less  than  two  generations,  they 
raise  themselves  from  the  lowest  stages  of  cultivation  to  scientific 

1  BihUotheca  mathematica  (2),  7,  1893,  p.  6. 

'' R.  Bonola,  Non-Euclidean  Geometry,  trand.  by  H.  S.  Carslaw,  Chicago,  1917, 
pp.  10-12. 


THE  ARABS  109 

efforts."    During  all  these  centuries,  astronomy  and  mathematics  in 
the  Orient  greatly  excel  these  sciences  in  the  Occident. 

Thus  far  we  have  spoken  only  of  the  Arabs  in  the  East.  Between 
the  Arabs  of  the  East  and  of  the  West,  which  were  under  separate  gov- 
ernments, there  generally  existed  considerable  political  animosity. 
In  consequence  of  this,  and  of  the  enormous  distance  between  the  two 
great  centres  of  learning,  Bagdad  and  Cordova,  there  was  less  scien- 
tific intercourse  among  them  than  might  be  expected  to  exist  between 
peoples  having  the  same  religion  and  written  language.  Thus  the 
course  of  science  in  Spain  was  quite  independent  of  that  in  Persia. 
While  wending  our  way  westward  to  Cordova,  we  must  stop  in  Egypt 
long  enough  to  observe  that  there,  too,  scientific  activity  was  re- 
kindled. Not  Alexandria,  but  Cairo  with  its  library  and  observatory, 
was  now  the  home  of  learning.  Foremost  among  her  scientists  ranked 
Ibn  Junos  (died  looS),  a  contemporary  of  Abu'I-Wefa.  He  solved 
some  difficult  problems  in  spherical  trigonometry.  Another  Egyptian 
astronomer  was  Ibn  Al-Haitam  (died  103S),  who  computed  the  vol- 
umes of  paraboloids  formed  by  revolving  a  parabola  about  any  diam- 
eter or  any  ordinate;  he  used  the  method  of  exhaustion  and  gave  the 
four  summation  formulas  for  the  first  four  powers  of  the  natural  num- 
bers.^ Travelling  westward,  we  meet  in  JVIorocco  Abu'l  Hasan  Ali, 
whose  treatise  "on  astronomical  instruments"  discloses  a  thorough 
knowledge  of  the  Conies  of  ApoUonius.  Arriving  finally  in  Spain 
at  the  capital,  Cordova,  we  are  struck  by  the  magnificent  splendor  of 
her  architecture.  At  this  renowned  seat  of  learning,  schools  and  li- 
braries were  founded  during  the  tenth  century. 

Little  is  known  of  the  progress  of  mathematics  in  Spain.  The  ear- 
liest name  that  has  come  down  to  us  is  Al-Majriti  (died  1007),  the 
author  of  a  mystic  paper  on  "amicable  numbers."  His  pupils  founded 
schools  at  Cordova,  Uania,  and  Granada.  But  the  only  great  astron- 
omer among  the  Saracens  in  Spain  is  Jabir  ibn  Aflah  of  Sevilla,  fre- 
quently called  Geber.  He  lived  in  the  second  half  of  the  eleventh  cen- 
tury. It  was  formerly  believed  that  he  was  the  inventor  of  algebra, 
and  that  the  word  algebra  came  from  "Jabir"  or  "  Geber."  He  ranks 
among  the  most  eminent  astronomers  of  this  time,  but,  hke  so  many 
of  his  contemporaries,  his  writings  contain  a  great  deal  of  mysticism. 
His  chief  work  is  an  astronomy  in  nine  books,  of  which  the  first  is  de- 
voted to  trigonometry.  In  his  treatment  of  spherical  trigonometry, 
he  exercises  great  independence  of  thought.  He  makes  war  against 
the  time-honored  procedure  adopted  by  Ptolemy  of  applying  "the 
rule  of  six  quantities,"  and  gives  a  new  way  of  his  own,  based  on  the 
"  rule  of  four  quantities."  This  is:  If  PPi  and  QQ\  be  two  arcs  of  great 
circles  intersecting  in  A,  and  if  PQ  and  P\Q\  be  arcs  of  great  circles 
drawn  perpendicular  to  QQi,  then  we  have  the  proportion 
sin  AP  :  sin  PQ  =  sm  APx  :  sin  PiQi. 
'H.  Suter  in  Bibliolheca  tnaihematka,  3.  S.,  Vol.  12,  1911-12,  pp.  320-322. 


no  A  HISTORY  OF  MATHEMATICS 

From  this  he  derives  the  formulas  for  spherical  right  triangles.  This 
sine-formula  was  probably  known  before  this  to  Tabit  ibn  Korra  and 
others.^  To  the  four  fundamental  formulas  already  given  by  Ptolemy, 
he  added  a  fifth,  discovered  by  himself.  If  a,  b,  c,  be  the  sides,  and 
A,  B,  C,  the  angles  of  a  spherical  triangle,  right-angled  at  A,  then 
cos  ^  =  cos  b  sin  C  This  is  frequently  called  "Geber's  Theorem." 
Radical  and  bold  as  were  his  innovations  in  spherical  trigonometry, 
in  plane  trigonometry  he  followed  slavishly  the  old  beaten  path  of 
the  Greeks.  Not  even  did  he  adopt  the  Indian  "sine"  and  "cosine," 
but  still  used  the  Greek  "chord  of  double  the  angle."  So  painful  was 
the  departure  from  old  ideas,  even  to  an  independent  Arab! 

It  is  a  remarkable  fact  that  among  the  early  Arabs  no  trace  what- 
ever of  the  use  of  the  abacus  can  be  discovered.  At  the  close  of  the 
thirteenth  century,  for  the  first  time,  do  we  find  an  Arabic  writer,  Ibn 
Albaima,  who  uses  processes  which  are  a  mixture  of  abacal  and  Hindu 
computation.  Ibn  Albanna  lived  in  Bugia,  an  African  seaport,  and  it 
is  plain  that  he  came  under  European  influences  and  thence  got  a 
knowledge  of  the  abacus.  Ibn  Albanna  and  Abraham  ibn  Esra  be- 
fore him,  solved  equations  of  the  first  degree  by  the  rule  of  "double 
false  position."  After  Ibn  Albanna  we  find  it  used  by  Al-Kalsadi 
and  Beha-Eddin  (1547-1622).^  If  ax-\-b  =  o,  let  m  and  n  be  any  two 
numbers  ("double  false  position"),  let  also  ain-\-b  =  M,  an-\-b  =  N, 
then  X  =  {nM  -  mN)  ^  {M  -  N) . 

Of  interest  is  the  approximate  solution  of  the  cubic  x^-\-Q  =  Px, 
which  grew  out  of  the  computation  of  x  =  sin  1°.  The  method  is 
shown  only  in  this  one  numerical  example.  It  is  given  in  Mir  am 
Chelebi  in  1498,  in  his  annotations  of  certain  Arabic  astronomical 
tables.  The  solution  is  attributed  to  Atabeddin  Jamshid.^  Write 
x={Q-\-x^)-^'P.  If  <3^P  =  a+i?-^P,  then  a  is  the  first  approxima- 
tion, X  being  snail.  We  have  Q  =  aP-\-R,  and  consequently  x  =  a-\- 
{R-\-a'^)-^P  =  a-\-b-\-S-^P^  say.  Then  a-\-b  is  the  second  approxima- 
tion. We  have  R  =  bP-\-S-a^  and  Q={a^-b)P^-S-a^.  Hence 
x  =  a+b-\-{S-a'^-\-{a+bf^P  =  a-\-b-{-c-\-T-^P,  say.  Here  a-\-b 
+c  is  the  third  approximation,  and  so  on.  In  general,  the  amount  of 
computation  is  considerable,  though  for  finding  x=sin  1°  the  method 
answered  very  well.  This  example  is  the  only  known  approximate 
arithmetical  solution  of  an  affected  equation  due  to  Arabic  writers. 
Nearly  three  centuries  before  this,  the  Italian,  Leonardo  of  Pisa, 
carried  the  solution  of  a  cubic  to  a  high  degree  of  approximation,  but 
without  disclosing  his  method. 

The  latest  prominent  Spanish-Arabic  scholar  was  Al-Kalsadi  of 
Granada,  who  died  in  i486.  He  wrote  the  Raising  of  the  Veil  of  the 
Science  of  Gubar.    The  word  "gubar"  meant  originally  "dust"  and 

1  See  Bibliotheca  mathematica ,  2  S.,  Vol.  7,  1893,  p.  7. 

2  L.  Matthiessen,  GrtmdzUge  d.  Antiken  u.  modernm  Algebra,  Leipzig,  1878,  p.  275. 
*  See  Cantofj  op.  cit.  Vol.  I,  3rd  Ed.,  1907,  p.  782. 


THE  ARABS  itt 

stands  here  for  written  arithmetic  with  numerals,  in  contrast  to  men- 
tal arithmetic.  In  addition,  subtraction  and  multiplication,  the 
result  is  written  above  the  other  figures.  The  square  root  was  indi- 
cated by  the  initial  Arabic  letter  of  the  word  "  jidre,"  meaning  "root," 
particularly  "square  root."  He  had  symbols  for  the  unknown  and 
had,  in  fact,  a  considerable  amount  of  algebraic  symbolism.  His 
approximation  for  the  square  root  Va-+6,  namely  (4a^-{- ;^ab) /  (4^-+ 
b),  is  believed  by  S.  Giinther  to  disclose  a  method  of  continued  frac- 
tions, without  our  modern  notation,  since  (4a^-\-T,ab)/(4a--\-b)  = 
a-{-b/{2a-\-b'2ii)-  Al-Kalsadi's  work  excels  other  Arabic  works  in 
the  amount  of  algebraic  symbolism  used.  Arabic  algebra  before  him 
contained  much  less  symbolism  then  Hindu  algebra.  With  Nessel- 
mann^  we  divide  algebras,  with  respect  to  notation,  into  three  classes: 
(i)  Rhetorical  algebras,  in  which  no  symbols  are  used,  everything 
being  written  out  in  words,  (2)  Syncopated  algebras,  in  which,  as  in 
the  first  class,  everything  is  WTitten  out  in  words,  except  that  abbrevia- 
tions are  used  for  certain  frequently  recurring  operations  and  ideas, 
(3)  Symbolic  alj^ebras,  in  which  all  forms  and  operations  are  repre- 
sented by  a  fully  developed  algebraic  symbolism,  as  for  example, 
x--!- 100:4-7.  According  to  this  classification,  Arabic  works  (excepting 
those  of  the  later  western  Arabs),  the  Greek  works  of  lamblichus  and 
Thymaridas,  and  the  works  of  the  early  Italian  writers  and  of  Regio- 
montanus  are  rhetorical  in  form;  the  works  of  the  later  v.-estern  Arabs, 
of  Diophantus  and  of  the  later  European  writers  down  to  about  the 
middle  of  the  seventeenth  century  (excepting  Vieta's  and  Oughtred's) 
are  syncopated  in  form;  the  Hindu  works  and  those  of  Meta  and 
Oughtred,  and  of  the  Europeans  since  the  middle  of  the  seventeenth 
century,  are  symbolic  in  form.  It  is  thus  seen  that  the  western  Arabs 
took  an  advanced  position  in  matters  of  algebraic  notation,  and  were 
inferior  to  none  of  their  predecessors  or  contemporaries,  except  the 
Hindus. 

In  the  year  in  which  Columbus  discovered  America,  the  Aloors 
lost  their  last  foot-hold  on  Spanish  soil;  the  productive  period  of 
Arabic  science  was  passed. 

We  have  witnessed  a  laudable  intellectual  activity  among  the 
Arabs.  They  had  the  good  fortune  to  possess  rulers  who,  by  their 
munificence,  furthered  scientific  research.  At  the  courts  of  the  ca- 
hphs,  scientists  were  supplied  with  libraries  and  obsers^atories.  A 
large  number  of  astronomical  and  mathematical  works  were  written 
by  Arabic  authors.  It  has  been  said  that  the  Arabs  were  learned, 
but  not  original.  With  our  present  knowledge  of  their  work,  this 
dictum  needs  revision;  they  have  to  their  credit  several  substantial 
accomplishments.  They  solved  cubic  equations  by  geometric  con- 
struction, perfected  trigonometry  to  a  marked  degree  and  made  nu- 

'  G.  H.  F.  Nesselmann,  Die  Algebra  dcr  Cricchen,  Berlin,  1842,  pp.  301-306. 


112  A  HISTORY  OF  MATHEMATICS 

merous  smaller  advances  all  along  the  line  of  mathematics,  physics 
and  astronomy.  Not  least  of  their  services  to  science  consists  in  this, 
that  they  adopted  the  learning  of  Greece  and  India,  and  kept  what 
they  received  with  care.  When  the  love  for  science  began  to  grow 
in  the  Occident,  they  transmitted  to  the  Europeans  the  valuable  treas- 
ures of  antiquity.  Thus  a  Semitic  race  was,  during  the  Dark  Ages, 
the  custodian  of  the  Aryan  intellectual  possessions. 


EUROPE  DURING  THE  MTJDl  h  AGES 

With  the  third  century  after  Christ  begins  an  era  of  migration  of 
nations  in  Europe.  The  powerful  Goths  quit  their  swamps  and  forests 
in  the  North  and  sweep  onward  in  steady  southwestern  current,  dis- 
lodging the  Vandals,  Sueves,  and  Burgundians,  crossing  the  Roman 
territory,  and  stopping  and  recoiling  only  when  reaching  the  shores 
of  the  Mediterranean.  From  the  Ural  Mountains  wild  hordes  sweep 
down  on  the  Danube.  The  Roman  Empire  falls  to  pieces,  and  the 
Dark  Ages  begin.  But  dark  though  they  seem,  they  are  the  germinat- 
ing season  of  the  institutions  and  nations  of  modern  Europe.  The 
Teutonic  element,  partly  pure,  partly  intermixed  with  the  Celtic  and 
Latin,  produces  that  strong  and  luxuriant  growth,  the  modern  civili- 
zation of  Europe.  Almost  all  the  various  nations  of  Europe  belong 
to  the  Aryan  stock.  As  the  Greeks  and  the  Hindus — both  Aryan  races 
— were  the  great  thinkers  of  antiquity,  so  the  nations  north  of  the  Alps 
and  Italy  became  the  great  intellectual  leaders  of  modern  times. 

Introduction  of  Roman  Mathematics 

We  shall  now  consider  how  these  as  yet  barbaric  nations  of  the 
North  gradually  came  in  possession  of  the  intellectual  treasures  of 
antiquity.  With  the  spread  of  Christianity  the  Latin  language  was 
introduced  not  only  in  ecclesiastical  but  also  in  scientific  and  all  im- 
portant worldly  transactions.  Naturally  the  science  of  the  Middle 
Ages  was  drawn  largely  from  Latin  sources.  In  fact,  during  the  earlier 
of  these  ages  Roman  authors  were  the  only  ones  read  in  the  Occident. 
Though  Greek  was  not  wholly  unknown,  yet  before  the  thirteenth 
century  not  a  single  Greek  scientific  work  had  been  read  or  translated 
into  Latin.  Meagre  indeed  was  the  science  which  could  be  gotten 
from  Roman  writers,  and  we  must  wait  several  centuries  before  any 
substantial  progress  is  made  in  mathematics. 

After  the  time  of  Boethius  and  Cassiodorius  mathematical  activity 
in  Italy  died  out.  The  first  slender  blossom  of  science  among  tribes 
that  came  from  the  North  was  an  encyclopaedia  entitled  Origenes, 
written  by  Isidorus  (died  636  as  bishop  of  Seville).  This  work  is 
modelled  after  the  Roman  encyclopaedias  of  Martianus  Capella  of 
Carthage  and  of  Cassiodorius.  Part  of  it  is  devoted  to  the  quadrivium, 
arithmetic,  music,  geometry,  and  astronomy.  He  gives  definitions 
and  grammatical  explications  of  technical  terms,  but  does  not  de- 
scribe the  modes  of  computation  then  in  vogue.  After  Isidorus  there 
follows  a  century  of  darkness  which  is  at  last  dissipated  by  the  appear- 

113 


114  A  HISTORY  OF  MATHEMATICS 

ance  of  Bede  the  Venerable  (672-735),  the  most  learned  man  of  his 
time.  He  was  a  native  of  VVearmouth,  in  England.  His  works  con- 
tain treatises  on  the  Computus,  or  the  computation  of  Easter-time, 
and  on  finger-reckoning.  It  appears  that  a  fmger-symbohsm  was  then 
widely  used  for  calculation.  The  correct  determination  of  the  time 
of  Easter  was  a  problem  which  in  those  days  greatly  agitated  the 
Church.  It  became  desirable  to  have  at  least  one  monk  at  each  mon- 
astery who  could  determine  the  day  of  religious  festivals  and  could 
compute  the  calendar.  Such  determinations  required  some  knowledge 
of  arithmetic.  Hence  we  find  that  the  art  of  calculating  always  found 
some  Uttle  corner  in  the  curriculum  for  the  education  of  monks. 

The  year  in  which  Bede  died  is  also  the  year  in  which  Alcuin  (735- 
804)  was  born.  Alcuin  was  educated  in  Ireland,  and  was  called  to  the 
court  of  Charlemagne  to  direct  the  progress  of  education  in  the  great 
Frankish  Empire.  Charlemagne  was  a  great  patron  of  learning  and 
of  learned  men.  In  the  great  sees  and  monasteries  he  founded  schools 
in  which  were  taught  the  psalms,  writing,  singing,  computation  {com- 
putus), and  grammar.  By  computus  was  here  meant,  probably,  not 
merely  the  determination  of  Easter-time,  but  the  art  of  computation 
in  general.  Exactly  what  modes  of  reckoning  were  then  employed 
we  have  no  means  of  knowing.  It  is  not  likely  that  Alcuin  was  familiar 
with  the  apices  of  Boethius  or  with  the  Roman  method  of  reckoning 
on  the  abacus.  He  belongs  to  that  long  list  of  scholars  who  dragged 
the  theory  of  numbers  into  theology.  Thus  the  number  of  beings 
created  by  God,  who  created  all  things  well,  is  6,  because  6  is  a  perfect 
number  (the  sum  of  its  divisors  being  iH-2+3  =  6);  8,  on  the  other 
hand,  is  an  imperfect  number  (i+2-|-4<8);  hence  the  second  origin 
of  mankind  emanated  from  the  number  8,  which  is  the  number  of  souls 
said  to  have  been  in  Noah's  ark. 

There  is  a  collection  of  "Problems  for  Quickening  the  Mind"  {prop- 
ositiones  ad  acucndos  iuvenes),  which  are  certainly  as  old  as  1000  a.  d. 
and  possibly  older.  Cantor  is  of  the  opinion  that  they  were  written 
much  earlier  and  by  Alcuin.  The  following  is  a  specimen  of  these 
"Problems":  A  dog  chasing  a  rabbit,  which  has  a  start  of  150  feet, 
jumps  9  feet  every  time  the  rabbit  jumps  7.  In  order  to  determine  in 
how  many  leaps  the  dog  overtakes  the  rabbit,  150  is  to  be  divided  by  2. 
In  this  collection  of  problems,  the  areas  of  triangular  and  quadrangular 
pieces  of  land  are  found  l^y  the  same  formulas  of  approximation  as 
those  used  by  the  Egyptians  and  given  by  Boethius  in  his  geometry. 
An  old  problem  is  the  "cistern-problem"  (given  the  time  in  which 
several  pipes  can  fill  a  cistern  singly,  to  find  the  time  in  which  they 
fill  it  jointly),  which  has  been  found  previously  in  Heron,  in  the  Greek 
Anthology,  and  in  Hindu  works.  Many  of  the  problems  shew  that 
the  collection  was  compiled  chiefly  from  Roman  sources.  The  prob- 
lem which,  on  account  of  its  uniqueness,  gives  the  most  positive  testi- 
mony regarding  the  Roman  origin  is  that  on  the  interpretation  of  a 


INTRODUCTION  OF  ROMAN  MATHEMATICS        115 

will  in  a  case  where  twins  are  born.  The  problem  is  identical  with  the 
Roman,  except  that  different  ratios  are  chosen.  Of  the  exercises  for 
recreation,  we  mention  the  one  of  the  wolf,  goat,  and  cabbage,  to  be 
rowed  across  a  river  in  a  boat  holding  only  one  besides  the  ferry-man. 
Query:  How  must  he  carry  them  across  so  that  the  goat  shall  not  eat 
the  cabbage,  nor  the  wolf  the  goat?  ^  The  solutions  of  the  "problems 
for  quickening  the  mind"  require  no  further  knowledge  than  the  recol- 
lection of  some  few  formulas  used  in  surveying,  the  ability  to  solve 
linear  equations  and  to  perform  the  four  fundamental  operations  with 
integers.  Extraction  of  roots  was  nowhere  demanded ;  fractions  hardly 
ever  occur.  - 

The  great  empire  of  Charlemagne  tottered  and  fell  almost  imme- 
diately after  his  death.  War  and  confusion  ensued.  Scientific  pur- 
suits were  abandoned,  not  to  be  resumed  until  the  close  of  the  tenth 
century,  when  under  Saxon  rule  in  Germany  and  Capetian  in  France, 
more  peaceful  times  began.  The  thick  gloom  of  ignorance  commenced 
to  disappear.  The  zeal  with  which  the  study  of  mathematics  was  now- 
taken  up  by  the  monks  is  due  principally  to  the  energy  and  influence 
of  one  man, — Gerbert.  He  was  born  in  Aurillac  in  Auvergne.  After 
receiving  a  monastic  education,  he  engaged  in  study,  chiefly  of  mathe- 
matics, in  Spain.  On  his  return  he  taught  school  at  Rheims  for  ten 
years  and  became  distinguished  for  his  profound  scholarship.  By 
King  Otto  I,  and  his  successors  Gerbert  was  held  in  highest  esteem. 
He  was  elected  bishop  of  Rheims,  then  of  Ravenna,  and  finally  was 
made  Pope  under  the  name  of  Sylvester  II,  by  his  former  Emperor 
Otho  III.  He  died  in  1003,  after  a  life  intricately  involved  in  many 
political  and  ecclesiastical  quarrels.  Such  was  the  career  of  the  great- 
est mathematician  of  the  tenth  century  in  Europe.  By  his  contem- 
poraries his  mathematical  knowledge  was  considered  wonderful. 
Many  even  accused  him  of  criminal  intercourse  with  evil  spirits. 

Gerbert  enlarged  the  stock  of  his  knowledge  by  procuring  copies 
of  rare  books.  Thus  in  Mantua  he  found  the  geometry  of  Boethius. 
Though  this  is  of  small  scientific  value,  yet  it  is  of  great  importance 
in  history.  It  was  at  that  time  the  principal  book  from  which  Euro- 
pean scholars  could  learn  the  elements  of  geometry.  Gerbert  studied 
it  with  zeal,  and  is  generally  believed  himself  to  be  the  author  of  a  ge- 
ometry. H.  Weissenborn  denied  his  authorship,  and  claimed  that  the 
book  in  question  consists  of  three  parts  which  cannot  come  from  one 
and  the  same  author.  More  recent  study  favors  the  conclusion  that 
Gerbert  is  the  author  and  that  he  compiled  it  from  different  sources.'^ 
This  geometry  contains  little  more  than  the  one  of  Boethius,  but  the 
fact  that  occasional  errors  in  the  latter  are  herein  corrected  shows  that 

'  S.  Giinther,  Geschichte  des  mathcm.  Unterrichls  im  deutschen  Mittelalter.  Berlin, 
1887,  p.  32- 

*M.  Cantor,  op.  cit.,  Vol.  I,  3.  Aufl.,  1907,  p.  S39. 

'  S.  Giinther,  Geschichte  der  Mathematik,  i.  Teil,  Leipzig,  1908,  p.  249. 


ii6  A  HISTORY  OF  MATHEMATICS 

the  author  had  mastered  the  subject,  "The  first  mathematical  paper 
of  the  Middle  Ages  which  deserves  th's  name,"  says  Hankel,  "is  a 
letter  of  Gerbert  to  Adalbold,  bishop  of  Utrecht,"  in  which  is  explained 
the  reason  why  the  area  of  a  triangle,  obtained  "geometrically"  by 
taking  the  product  of  the  base  by  half  its  altitude,  differs  from  the 
area  calculated  "arithmetically,"  according  to  the  formula  |a(a+i), 
used  by  surveyors,  where  a  stands  for  a  side  of  an  equilateral  triangle. 
He  gives  the  correct  explanation  that  in  the  latter  formula  all  the 
small  squares,  in  which  the  triangle  is  supposed  to  be  divided,  are 
counted  in  wholly,  even  though  parts  of  them  project  beyond  it. 
D.  E.  Smith  ^  calls  attention  to  a  great  medieval  number  game,  called 
rithmomachia,  claimed  by  some  to  be  of  Greek  origin.  It  was  played 
as  late  as  the  sixteenth  century.  It  called  for  considerable  arithmeti- 
cal ability,  and  was  known  to  Gerbert,  Oronce  Fine,  Thomas  Brad- 
wardine  and  others.  A  board  resembling  a  chess  board  was  used.  Re- 
lations like  8i  =  72-|-|  of  72,  42  =  36+!  of  36  were  involved. 

Gerbert  made  a  careful  study  of  the  arithmetical  works  of  Boethius. 
He  himself  pubUshed  the  first,  perhaps  both,  of  the  following  two 
works, — A  Small  Book  on  the  Division  of  Numbers,  and  Rule  of  Compu- 
tation on  the  Abacus.  They  give  an  insight  into  the  methods  of  calcu- 
lation practised  in  Europe  before  the  introduction  of  the  Hindu  nu- 
merals. Gerbert  used  the  abacus,  which  was  probably  unknown  to 
Alcuin.  Bernelinus,  a  pupil  of  Gerbert,  describes  it  as  consisting  of 
a  smooth  board  upon  which  geometricians  were  accustomed  to  strew 
blue  sand,  and  then  to  draw  their  diagrams.  For  arithmetical  pur- 
poses the  board  was  divided  into  30  columns,  of  which  3  were  reserved 
for  fractions,  while  the  remaining  27  were  divided  into  groups  with 
3  columns  in  each.  In  every  group  the  columns  were  marked  respec- 
tively by  the  letters  C  (centum),  D  (decern),  and  S  {singularis)  or 
M  (monas).  Bernelinus  gives  the  nine  numerals  used,  which  are  the 
apices  of  Boethius,  and  then  remarks  that  the  Greek  letters  may  be 
used  in  their  place.  By  the  use  of  these  columns  any  number  can  be 
written  without  introducing  a  zero,  and  all  operations  in  arithmetic 
can  be  performed  in  the  same  way  as  we  execute  ours  without  the  col- 
umns, but  with  the  symbol  for  zero.  Indeed,  the  methods  of  adding, 
subtracting,  and  multiplying  in  vogue  among  the  abacists  agree  sub- 
stantially with  those  of  to-day.  But  in  a  division  there  is  very  great 
difference.  The  early  rules  for  division  appear  to  have  been  framed 
to  satisfy  the  following  three  conditions:  (i)  The  use  of  the  multipli- 
cation table  shall  be  restricted  as  far  as  possible;  at  least,  it  shall  never 
be  required  to  multiply  mentally  a  figure  of  two  digits  by  another  of 
one  digit.  (2)  Subtractions  shall  be  avoided  as  much  as  possible  and 
replaced  by  additions.  (3)  The  operation  shall  proceed  in  a  purely 
mechanical  way,  without  requiring  trials.^  That  it  should  be  neces- 
sary to  make  such  conditions  seems  strange  to  us;  but  it  must  be  re_ 

'  Am.  Math.  Monthly,  Vol.  28,  191 1,  pp.  73-80.     ^  jj.  Hankel,  op.  ciL,  p.  318. 


INTRODUCTION  OF  ROMAN  MATHEMATICS        117 

remembered  that  the  monks  of  the  Middle  Ages  did  not  attend  school 
during  childhood  and  learn  the  multiplication  table  while  the  memory 
was  fresh.  Gerbert's  rules  for  division  are  the  oldest  extant.  They 
are  so  brief  as  to  be  very  obscure  to  the  uninitiated.  They  were  prob- 
ably intended  simply  to  aid  the  memory  by  calling  to  mind  the  suc- 
cessive steps  in  the  work.  In  later  manuscripts  they  are  stated  more 
fully.  In  dividing  any  number  by  another  of  one  digit,  say  668  by  6, 
the  divisor  was  first  increased  to  10  by  adding  4.  The  process  is  ex- 
hibited in  the  adjoining  figure.^  As  it  continues,  we  must  imagine  the 
digits  which  are  crossed  out,  to  be  erased  and  then  replaced  by  the 
ones  beneath.  It  is  as  follows:  600-i- 10  =  60,  but,  to  rectify  the  error, 
4X60,  or  240,  must  be  added;  200-^-10=20,  but  4X20,  or  80,  must 
be  added.  \Ve  now  write  for  60-I-40+80,  its  sum  180,  and  continue 
thus:  100-^10=10;  the  correction  necessary  is  4X10,  or  40,  which, 
added  to  80,  gives  120.  Now  :ioo-^  10=  10,  and  the  correction  4X  10, 
together  with  the  20,  gives  60.  Proceeding  as  before,  6o-r- 10  =  6;  the 
correction  is  4X  6  =  24.  Now  20-^- 10=  2,  the  correction  being  4X  2  =  8. 
In  the  column  of  units  we  have  now  8+4+S,  or  20.  As  before,  20-7- 
10=  2;  the  correction  is  2X4=  8,  which  is  not  divisible  by  10,  but  only 
by  6,  giving  the  quotient  i  and  the  remainder  2.  All  the  partial  quo- 
tients taken  together  give  60+20+10+10+6+2+2+1  =  111,  and 
the  remainder  2. 

Similar  but  more  compHcated,  is  the  process  when  the  divisor  con- 
tains two  or  more  digits.  Were  the  divisor  27,  then  the  next  higher 
multiple  of  10,  or  30,  would  be  taken  for  the  divisor,  but  corrections 
would  be  required  for  the  3.  He  who  has  the  patience  to  carry  such 
a  division  through  to  the  end,  will  understand  why  it  has  been  said  of 
Gerbert  that  "Regulas  dedit,  quK  a  sudantibus  abacistis  vix  intelli- 
guntur."  He  will  also  perceive  why  the  Arabic  method  of  division, 
when  first  introduced,  was  called  the  divisio  aurea,  but  the  one  on  the 
abacus,  the  divisio  Jerrea. 

In  his  book  on  the  abacus,  Bernelinus  devotes  a  chapter  to  fractions. 
These  are,  of  course,  the  duodecimals,  first  used  by  the  Romans.  For 
want  of  a  suitable  notation,  calculation  with  them  was  exceedingly 
difficult.  It  would  be  so  even  to  us,  were  we  accustomed,  like  the 
early  abacists,  to  express  them,  not  by  a  numerator  or  denominator, 
but  by  the  application  of  names,  such  as  uncia  for  i\,  quincunx  for  i\. 
dodrans  for  i\. 

In  the  tenth  century,  Gerbert  was  the  central  figure  among  the 
learned.  In  his  time  the  Occident  came  into  secure  possession  of  all 
mathematical  knowledge  of  the  Romans.  During  the  eleventh  cen- 
tury it  was  studied  assiduously.  Though  numerous  works  were 
written  on  arithmetic  and  geometry,  mathematical  knowledge  in  the 
Occident  was  still  very  insignificant.  Scanty  indeed  were  the  mathe- 
matical treasures  obtained  from  Roman  sources. 

1  M.  Cantor,  op.  cil.,  Vol.  I,  3.  Aufl.,  1907,  p.  882. 


ii8  A  HISTORY  OF  MATHEMATICS 

Translation  oj  Arabic  Manuscripts 

By  his  great  erudition  and  phenomenal  activity,  Gerbert  infused 
new  Ufe  into  the  study  not  only  of  mathematics,  but  also  of  philosophy. 
Pupils  from  France,  Germany,  and  Italy  gathered  at  Rheims  to  enjoy 
his  instruction.  When  they  themselves  became  teachers,  they  taught 
of  course  not  only  the  use  of  the  abacus  and  geometry,  but  also  what 
they  had  learned  of  the  philosophy  of  Aristotle.  His  philosophy  was 
known,  at  first,  only  through  the  writings  of  Boethius.  But  the  grow- 
ing enthusiasm  for  it  created  a  demand  for  his  complete  works.  Greek 
texts  were  wanting.  But  the  Latins  heard  that  the  Arabs,  too,  were 
great  admirers  of  Peripatetism,  and  that  they  possessed  translations 
of  Aristotle's  works  and  commentaries  thereon.  This  led  them  finally 
to  search  for  and  translate  Arabic  manuscripts.  During  this  search, 
mathematical  works  also  came  to  their  notice,  and  were  translated 
into  Latin.  Though  some  few  unimportant  works  may  have  been 
translated  earher,  yet  the  period  of  greatest  activity  began  about  iioo. 
The  zeal  displayed  in  acquiring  the  Mohammedan  treasures  of  knowl- 
edge excelled  even  that  of  the  Arabs  themselves,  when,  in  the  eighth 
century,  they  plundered  the  rich  coffers  of  Greek  and  Hindu  science. 

Among  the  earliest  scholars  engaged  in  translating  manuscripts  into 
Latin  was  Athelard  of  Bath.  The  period  of  his  activity  is  the  first 
quarter  of  the  twelfth  century.  He  travelled  extensively  in  Asia 
Minor,  Egypt,  perhaps  also  in  Spain,  and  braved  a  thousand  perils, 
that  he  might  acquire  the  language  and  science  of  the  Mohammedans. 
He  made  one  of  the  earliest  translations,  from  the  Arabic,  of  Euclid's 
Elements.  He  translated  the  astronomical  tables  of  Al-Khowarizmi. 
In  1857,  a  manuscript  was  found  in  the  library  at  Cambridge,  which 
proved  to  be  the  arithmetic  by  Al-Khowarizmi  in  Latin.  This  trans- 
lation also  is  very  probably  due  to  Athelard. 

At  about  the  same  time  flourished  Plato  of  Tivoli  or  Plato  Tihurtinus. 
He  effected  a  translation  of  the  astronomy  of  Al-Battani  and  of  the 
Sphcerica  of  Theodosius. 

About  the  middle  of  the  twelfth  century  there  was  a  group  of  Chris- 
tian scholars  busily  at  work  at  Toledo,  under  the  leadership  of  Ray- 
mond, then  archbishop  of  Toledo.  Among  those  who  worked  under 
his  direction,  John  of  Seville  was  most  prominent.  He  translated 
works  chiefly  on  Aristotelian  philosophy.  Of  importance  to  us  is  a 
liher  alghoarisnii,  compiled  by  him  from  Arabic  authors.    The  rule  for 

a     c 
the  division  of  one  fraction  by  another  is  proved  as  follows:  j^j  = 

^-^— =— .  This  same  explanation  is  given  by  the  thirteenth  cen- 
hd     bd     be 

tury  German  writer,  Jordanus  Nemorarius.  On  comparing  works 
like  this  with  those  of  the  abacists,  v/e  notice  at  once  the  most  striking 
difference,  which  shows  that  the  two  parties  drew  from  independent 


TRANSLATION  OF  ARABIC  MANUSCRIPTS  itq 

sources.  It  is  argued  by  some  that  Gerbert  got  his  apices  and  his  arith- 
metical knowledge,  not  from  Boethius,  but  from  the  Arabs  in  Spain, 
and  that  part  or  the  whole  of  the  geometry  of  Boethius  is  a  forgery, 
dating  from  the  time  of  Gerbert.  If  this  were  the  case,  then  the  writ- 
ings of  Gerbert  would  betray  Arabic  sources,  as  do  those  of  John  of 
Seville.  But  no  points  of  resemblance  are  found.  Gerbert  could  not 
have  learned  from  the  Arabs  the  use  of  the  abacus,  because  all  evidence 
we  have  goes  to  show  that  they  did  not  employ  it.  Nor  is  it  probable 
that  he  borrowed  from  the  Arabs  the  apices,  because  they  were  never 
used  in  Europe  except  on  the  abacus.  In  illustrating  an  example  in 
division,  mathematicians  of  the  tenth  and  eleventh  centuries  state 
an  example  in  Roman  numerals,  then  draw  an  abacus  and  insert  in  it 
tlie  necessary  numbers  with  the  apices.  Hence  it  seems  probable  that 
the  abacus  and  apices  were  borrowed  from  the  same  source.  The 
contrast  between  authors  like  John  of  Seville,  drawing  from  Arabic 
works,  and  the  abacists,  consists  in  this,  that,  unlike  the  latter,  the 
former  mention  the  Hindus,  use  the  term  algorism,  calculate  with  the 
zero,  and  do  not  employ  the  abacus.  The  former  teach  the  extraction 
of  roots,  the  abacists  do  not;  they  teach  the  sexagesimal  fractions  used 
by  the  Arabs,  while  the  abacists  employ  the  duodecimals  of  the  Ro- 
mans.^ 

A  little  later  than  John  of  Seville  flourished  Gerard  of  Cremona  in 
Lombardy.  Being  desirous  to  gain  possession  of  the  Almagest,  he 
went  to  Toledo,  and  there,  in  1175,  translated  this  great  work  of  Ptol- 
emy. Inspired  by  the  richness  of  Mohammedan  literature,  he  gave 
himself  up  to  its  study.  He  translated  into  Latin  over  70  Arabic  works. 
Of  mathematical  treatises,  there  were  among  these,  besides  the  Al- 
magest, the  15  books  of  Euclid,  the  Splicer ica  of  Theodosius,  a  work  of 
Menelaus,  the  algebra  of  Al-Khowarizmi,  the  astronomy  of  Jabir  ibn 
Aflah,  and  others  less  important.  Through  Gerard  of  Cremona  the 
term  sinus  was  introduced  into  trigonometry.  Al-Khawarizmi's  al- 
gebra was  translated  also  by  Robert  of  Chester;  his  translation  prob- 
ably antedated  Cremona's. 

In  the  thirteenth  century,  the  zeal  for  the  acquisition  of  Arabic 
learning  continued.  Foremost  among  the  patrons  of  science  at  this 
time  ranked  Emperor  Frederick  II  of  Hohenstaufen  (died  1250). 
Through  frequent  contact  with  Mohammedan  scholars,  he  became 
familiar  with  Arabic  science.  He  employed  a  number  of  scholars  in 
translating  Arabic  manuscripts,  and  it  was  through  him  that  we  came 
in  possession  of  a  new  translation  of  the  Almagest.  Another  royal 
head  deserving  mention  as  a  zealous  promoter  of  Arabic  science  was 
Alfonso  X  of  Castile  (died  1284).  He  gathered  around  him  a  number 
of  Jewish  and  Christian  scholars,  who  translated  and  compiled  astro 
nomical  works  from  Arabic  sources.  Astronomical  tables  prepared 
by  two  Jews  spread  rapidly  in  the  Occident,  and  constituted  the  basis 
1  M.  Cantor,  op.  cit.,  Vol.  I,  3.  Aufl.,  1907,  p.  879,  chapter  40. 


120  A  HISTORY  OF  MATHEMATICS 

of  all  astronomical  calculation  till  the  sixteenth  century.  The  num- 
ber of  scholars  who  aided  in  transplanting  Arabic  science  upon  Chris- 
tian soil  was  large.  But  we  mention  only  one,  Giovanni  Campano  of 
Novara  (about  1260),  who  brought  out  a  new  translation  of  Euclid, 
which  drove  the  earlier  ones  from  the  field,  and  which  formed  the 
basis  of  the  printed  editions.  ^ 

At  the  middle  of  the  twelfth  century,  the  Occident  was  in  possession 
of  the  so-called  Arabic  notation.  At  the  close  of  the  century,  the 
Hindu  methods  of  calculation  began  to  supersede  the  cumbrous  meth- 
ods inherited  from  Rome.  Algebra,  with  its  rules  for  solving  linear 
and  quadratic  equations,  had  been  made  accessible  to  the  Latins.  The 
geometry  of  Euclid,  the  Sphfrrica  of  Theodosius,  the  astronomy  of 
Ptolemy,  and  other  works  were  now  accessible  in  the  Latin  tongue. 
Thus  a  great  amount  of  new  scientific  material  had  come  into  the 
hands  of  the  Christians.  The  talent  necessary  to  digest  this  hetero- 
geneous mass  of  knowledge  was  not  wanting.  The  figure  of  Leonardo 
of  Pisa  adorns  the  vestibule  of  the  thirteenth  century. 

It  is  important  to  notice  that  no  work  either  on  mathematics  or 
astronomy  was  translated  directly  from  the  Greek  previous  to  the 
fifteenth  century. 

The  First  Awakening  and  its  Sequel 

Thus  far,  France  and  the  British  Isles  have  been  the  headquarters 
of  mathematics  in  Christian  Europe.  But  at  the  beginning  of  the 
thirteenth  century  the  talent  and  activity  of  one  man  was  sufficient  to 
assign  the  mathematical  science  a  new  home  in  Italy.  This  man  was 
not  a  monk,  like  Bede,  Alcuin,  or  Gerbert,  but  a  layman  who  found 
time  for  scientific  study.  Leonardo  of  Pisa  is  the  man  to  whom  we 
owe  the  first  renaissance  of  mathematics  on  Christian  soil.  He  is  also 
called  Fibonacci,  i.e.  son  of  Bonaccio.  His  father  was  secretary  at  one  of 
the  numerous  factories  erected  on  the  south  and  east  coast  of  the  Med- 
iterranean by  the  enterprising  merchants  of  Pisa.  He  made  Leonardo, 
when  a  boy,  learn  the  use  of  the  abacus.  The  boy  acquired  a  strong 
taste  for  mathematics,  and,  in  later  years,  during  extensive  travels  in 
Egypt,  Syria,  Greece,  and  Sicily,  collected  from  the  various  peoples 
all  the  knowledge  he  could  get  on  this  subject.  Of  all  the  methods  of 
calculation,  he  found  the  Hindu  to  be  unquestionably  the  best.  Re- 
turning to  Pisa,  he  published,  in  1202,  his  great  work,  the  Liber  Abaci. 
A  revised  edition  of  this  appeared  in  1228.  This  work  contains  the 
knowledge  the  Arabs  possessed  in  arithmetic  and  algebra,  and  treats 
the  subject  in  a  free  and  independent  way.  This,  together  with  the 
other  books  of  Leonardo,  shows  that  he  was  not  merely  a  compiler, 
nor,  like  other  writers  of  the  Middle  Ages,  a  slavish  imitator  of  the 
form  in  which  the  subject  had  been  previously  presented.    The  extent 

1 H.  Hankel,  op.  cit.,  pp.  338,  339. 


THE  FIRST  AWAKENING  AND  ITS  SEQUEL         121 

of  his  originality  is  not  definitely  known,  since  the  sources  from  which 
he  drew  have  not  all  been  ascertained.  Karpinski  has  shown  that 
Leonardo  drew  extensively  from  Abu  Kamil's  algebra.  Leonardo's 
Praciica  geometric  is  partly  drawn  from  the  Liber  embadorum  of  Sav- 
asorda,  a  learned  Jew  of  Barcelona  and  a  co-worker  of  Plato  of 
TivoU. 

Leonardo  was  the  first  great  mathematician  to  advocate  the  adop- 
tion of  the  "Arabic  notation."  The  calculation  with  the  zero  was  the 
portion  of  Arabic  mathematics  earhest  adopted  by  the  Christians. 
The  minds  of  men  had  been  prepared  for  the  reception  of  this  by  the 
use  of  the  abacus  and  the  apices.  The  reckoning  with  columns  was 
gradually  abandoned,  and  the  very  word  abacus  changed  its  meaning 
and  became  a  synonym  for  algorism.  For  the  zero,  the  Latins  adopted 
the  name  zephirum,  from  the  Arabic  sifr  (sifra  =  empty);  hence  our 
English  word  cipher.  The  new  notation  was  accepted  readily  by  the 
enlightened  masses,  but,  at  first,  rejected  by  the  learned  circles.  The 
merchants  of  Italy  used  it  as  early  as  the  thirteenth  century,  while 
the  monks  in  the  monasteries  adhered  to  the  old  forms.  In  1299, 
nearly  100  years  after  the  publication  of  Leonardo's  Liber  Abaci,  the 
Florentine  merchants  were  forbidden  the  use  of  the  Arabic  numeral 
in  book-keeping,  and  ordered  either  to  employ  the  Roman  numerals 
or  to  write  the  numeral  adjectives  out  in  full.  This  decree  is  probably 
due  to  the  variety  of  forms  of  certain  digits  and  the  consequent  am- 
biguity, misunderstanding  and  fraud.  Some  interest  attaches  to  the 
earliest  dates  indicating  the  use  of  Hindu-Arabic  numerals  in  the  Oc- 
cident. Many  erroneous  or  doubtful  early  dates  have  been  given  by 
writers  inexperienced  in  the  reading  of  manuscripts  and  inscriptions. 
The  numerals  are  first  found  in  manuscripts  of  the  tenth  century,  but 
they  were  not  well  known  until  the  beginning  of  the  thirteenth  cen- 
tury.^ About  1275  they  began  to  be  widely  used.  The  earliest  Arabic 
manuscripts  containing  the  numerals  are  of  874  and  888  A.  d.  They 
appear  in  a  work  written  at  Shiraz  in  Persia  in  970  a.  d.  A  church- 
pillar  not  far  from  the  Jeremias  Monastery  in  Egypt  has  the  date  349 
A.  H.  (  =  961  A.  D.)  The  oldest  definitely  dated  European  manuscript 
known  to  contain  the  numerals  is  the  Codex  Vigilanus,  written  in  the 
Albelda  Cloister  in  Spain  in  976  A.  D.  The  nine  characters  without 
the  zero  are  given,  as  an  addition,  in  a  Spanish  copy  of  the  Origines 
by  Isidorus  of  Seville,  992  a.  d.  A  tenth  century  manuscript  with 
forms  difi"ering  materially  from  those  in  the  Codex  Vigilanus  was  found 
in  the  St.  Gall  manuscript  now  in  the  University  Library  at  Zurich. 
The  numerals  are  contained  in  a  Vatican  manuscript  of  1077,  a  Sicilian 
coin  of  1 138,  a  Regensburg  (Bavaria)  chronicle  of  1197.  The  earliest 
manuscript  in  French  giving  the  numerals  dates  about  1275.    In  the 

^  G.  F.  Hill,  The  Devrlopmenl  of  Arabic  Numerals  in  Europe,  Oxford,  IQ15,  p.  11. 
Our  dates  are  taken  from  this  hook  and  from  I).  E.  Smith  and  L.  C.  Karpinski's 
Hindu-Arabic  Mumcrah,  Boston  and  London,  191 1,  pp.  133-146. 


T22  A  HISTORY  OF  MATHEMATICS 

British  Museum  one  English  manuscript  is  of  about  1230-50,  another 
IS  of  1246.  The  earUest  undoubted  Arabic  numerals  on  a  gravestone 
are  at  Pforzheim  in  Baden  of  137 1  and  one  at  Ulm  of  1388.  The 
earliest  coins  dated  in  the  Arabic  numerals  are  as  follows:  Swiss  1424, 
Austrian  1484,  French  1485,  German  1489,  Scotch  1539,  English  1551. 
The  earUest  calendar  with  Arabic  figures  is  that  of  Kobel,  1518.  The 
forms  of  the  numerals  varied  considerably.  The  5  was  the  most 
freakish.    An  upright  7  was  rare  in  the  earlier  centuries. 

In  the  fifteenth  century  the  abacus  with  its  counters  ceased  to  be 
used  in  Spain  and  Italy.  In  France  it  was  used  later,  and  it  did  not 
disappear  in  England  and  Germany  before  the  middle  of  the  seven- 
teenth century.^  The  method  of  abacal  computation  is  found  in  the 
EngUsh  exchequer  for  the  last  time  in  1676.  In  the  reign  of  Henry  I 
the  exchequer  was  distinctly  organized  as  a  court  of  law,  but  the  finan- 
cial business  of  the  crown  was  also  carried  on  there.  The  term  "ex- 
chequer" is  derived  from  the  chequered  cloth  which  covered  the  table 
at  which  the  accounts  were  made  up.  Suppose  the  sheriff  was  sum- 
moned to  answer  for  the  full  annual  dues  "in  money  or  in  tallies." 
"The  liabilities  and  the  actual  payments  of  the  sheriff  were  balanced 
by  means  of  counters  placed  upon  the  squares  of  the  chequered  table 
those  on  the  one  side  of  the  table  representing  the  value  of  the  tallies, 
warrants  and  specie  presented  by  the  sheriff,  and  those  on  the  other 
the  amount  for  which  he  was  liable,"  so  that  it  was  easy  to  see  whether 
the  sheriff  had  met  his  obligations  or  not.  In  Tudor  times  "pen  and 
ink  dots"  took  the  place  of  counters.  These  dots  were  used  as  late  as 
1676.^  The  "tally"  upon  which  accounts  were  kept  was  a  peeled 
wooden  rod  split  in  such  a  way  as  to  divide  certain  notches  previously 
cut  in  it.  One  piece  of  the  tally  was  given  to  the  payer;  the  other  piece 
was  kept  by  the  exchequer.  The  transaction  could  be  verified  easily 
by  fitting  the  two  halves  together  and  noticing  whether  the  notches 
"  talHed"  or  nor.    Such  tallies  remained  in  use  as  late  as  1783. 

In  the  Winter's  Tale  (IV.  3),  Shakespeare  lets  the  clown  be  embar- 
rassed by  a  problem  which  he  could  not  do  without  counters.  lago  (in 
Othello,  i,  i)  expresses  his  contempt  for  Michael  Cassio,  "forsooth  a 
great  mathematician,"  by  caUing  him  a  "counter-caster."  ^  So  gen- 
eral, indeed,  says  Peacock,  appears  to  have  been  the  practice  of  this 
species  of  arithmetic,  that  its  rules  and  principles  form  an  essential 
part  of  the  arithmetical  treatises  of  that  day.  The  real  fact  seems  to 
be  that  the  old  methods  were  used  long  after  the  Hindu  numerals  were 

1  George  Peacock,  "Arithmetic"  in  the  Encydopcedia  of  Pure  Mathematics, 
London,  1847,  p.  408. 

2  Article  "Exchequer"  in  Palgrave's  Dictiottary  of  Political  Economy,  London, 
1894. 

*For  additional  infonnation,  consult  F.  P.  Barnard,  The  Casting-Counter  and 
the  Counting-Board,  Oxford,  1916.  He  gives  a  list  of  159  extracts  from  English 
inventories  referring  to  counting  boards  and  also  photographs  of  reckoning  tables 
at  Basel  and  Niirnberg,  of  reckoning  cloths  at  Munich,  etc.    . 


THE  FIRST  AWAKENING  AND  ITS  SEQUEL         123 

in  common  and  general  use.  With  such  dogged  persistency  does  man 
cling  to  the  old! 

The  Liber  Abaci  was,  for  centuries,  one  of  the  storehouses  from 
which  authors  got  material  for  works  on  arithmetic  and  algebra.  In 
it  are  set  forth  the  most  perfect  methods  of  calculation  with  integers 
and  fractions,  known  at  that  time;  the  square  and  cube  root  are  ex- 
plained, cube  root  nor  having  been  considered  in  the  Christian  Occi- 
dent before;  equations  of  the  first  and  second  degree  leading  to  prob- 
lems, either  determinate  or  indeterminate,  are  solved  by  the  methods 
of  "single"  or  "double  position,"  and  also  by  real  algebra.  He  recog- 
nized that  the  quadratic  x^'-^-c^bx  may  be  satisfied  by  two  values  of  x. 
He  took  no  cognizance  of  negative  and  imaginary  roots.  The  book 
contains  a  large  number  of  problems.  The  following  was  proposed  to 
Leonardo  of  Pisa  by  a  magister  in  Constantinople,  as  a  difiicult  prob- 
lem: If  A  gets  from  B  7  denare,  then  A's  sum  is  five-fold  B's;  if  B  gets 
from  A  5  denare,  then  B's  sum  is  seven-fold  A's.  How  much  has  each? 
The  Liber  Abaci  contains  another  problem,  which  is  of  historical  in- 
terest, because  it  was  given  with  some  variations  by  Ahmes,  3000  years 
earlier:  7  old  women  go  to  Rome;  each  woman  has  7  mules,  each  mule 
carries  7  sacks,  each  sack  contains  7  loaves,  with  each  loaf  are  7  knives, 
each  knife  is  put  up  in  7  sheaths.  What  is  the  sum  total  of  all  named? 
Ans.  137,256.-^  Following  the  practice  of  Arabic  and  of  Greek  and 
Egyptian  writers,  Leonardo  frequently  uses  unit  fractions.  This  was 
done  also  by  other  European  writers  of  the  Middle  Ages.  He  ex- 
plained how  to  resolve  a  fraction  into  the  sum  of  unit  fractions.  He 
was  one  of  the  first  to  separate  the  numerator  from  the  denominator 
by  a  fractional  line.  Before  his  time,  when  fractions  were  written  in 
Hindu-Arabic  numerals,  the  denominator  was  written  beneath  the 
numerator,  without  any  sign  of  separation. 

In  1220,  Leonardo  of  Pisa  published  his  Practica  Geometric^,  which 
contains  all  the  knowledge  of  geometry  and  trigonometry  transmitted 
to  him.  The  writings  of  Euclid  and  of  some  other  Greek  masters  were 
known  to  him,  either  from  Arabic  manuscripts  directly  or  from  the 
translations  made  by  his  countrymen,  Gerard  of  Cremona  and  Plato 
of  Tivoh.  As  previously  stated,  a  principal  source  of  his  geometrical 
knowledge  was  Plato  of  Tivolis'  translation  in  11 16,  from  the  Hebrew 
into  Latin,  of  the  Liber  embadorum  of  Abraham  Savasorda.-  Leo- 
nardo's Geometry  contains  an  elegant  geometrical  demonstration  of 
Heron's  formula  for  the  area  of  a  triangle,  as  a  function  of  its  three 
sides;  the  proof  resembles  Heron's.  Leonardo  treats  the  rich  material 
before  him  with  skill,  some  originality  and  Euclidean  rigor. 

Of  still  greater  interest  than  the  preceding  works  are  those  contain- 

'  M.  Cantor,  op.  cil.,  Vol.  IT,  2.  Aufl.,  1900,  p.  26.  See  a  problem  in  the  Ahmes 
Dapyrus  believed  to  be  of  the  same  type  as  this. 

^  See  M.  Curtze,  Urkunden  zur  Geschichle  der  Malhcmatik,  I  Thcil,  Leipzig,  1902, 
^-  5- 


124  A  HISTORY  OF  MATHEMATICS 

ing  Fibonacci's  more  original  investigations.  We  must  here  preface 
that  after  the  publication  of  the  Liber  Abaci,  Leonardo  was  presented 
by  the  astronomer  Dominicus  to  Emperor  Frederick  II  of  Hohen- 
staufen.  On  that  occasion,  John  of  Palermo,  an  imperial  notary, 
proposed  several  problems,  which  Leonardo  solved  promptly.  The 
first  (probably  an  old  familiar  problem  to  him)  was  to  find  a  number  x, 
such  that  X-+5  and  x^— 5  are  each  square  numbers.  The  answer  is 
^  =  3tt;  for  (3l\)--f-5=(4lV)^  (3r2)"-5=  (2t'2)''-  His  masterly  so- 
lution of  this  is  given  in  his  liber  quadratorum,  a  manuscript  which  was 
not  printed,  but  to  which  reference  is  made  in  the  second  edition  of 
his  Liber  Abaci.  The  problem  was  not  original  with  John  of  Palermo, 
since  the  Arabs  had  already  solved  similar  ones.  Some  parts  of  Leo- 
nardo's solution  may  have  been  borrowed  from  the  Arabs,  but  the 
method  which  he  employed  of  building  squares  by  the  summation  of 
odd  numbers  is  original  with  him. 

The  second  problem  proposed  to  Leonardo  at  the  famous  scientific 
tournament  which  accompanied  the  presentation  of  this  celebrated  al- 
gebraist to  that  great  patron  of  learning,  Emperor  Frederick  II,  was 
the  solving  of  the  equation  x^+  2x--\-  io.v=  20.  As  yet  cubic  equations 
had  not  been  solved  algebraically.  Instead  of  brooding  stubbornly 
over  this  knotty  problem,  and  after  many  failures  still  entertaining 
new  hopes  of  success,  he  changed  his  method  of  inquiry  and  showed 
by  clear  and  rigorous  demonstration  that  the  roots  of  this  equation 
could  not  be  represented  by  the  Euclidean  irrational  quantities,  or,  in 
other  words,  that  they  could  not  be  constructed  with  the  ruler  and 
compass  only.  He  contented  himself  with  finding  a  very  close  ap- 
proximation to  the  required  root.  His  work  on  this  cubic  is  found  in 
the  Flos,  together  with  the  solution  of  the  following  third  problem 
given  him  by  John  of  Palermo:  Three  men  possess  in  common  an  un- 
known sum  of  money  /;  the  share  of  the  first  is  -;  that  of  the  second,  - ; 
that  of  the  third,  -.  Desirous  of  depositing  the  sum  at  a  safer  place, 
each  takes  at  hazard  a  certain  amount;  the  first  takes  x,  but  deposits 
only  - ;  the  second  carries  y,  but  deposits  only  -;  the  third  takes  z,  and 

deposits  -.    Of  the  amount  deposited  each  one  must  receive  exactly  \, 

in  order  to  possess  his  share  of  the  whole  sum.  Find  x,  y,  z.  Leonardo 
shows  the  problem  to  be  indeterminate.  Assuming  7  for  the  sum 
drawn  by  each  from  the  deposit,  he  finds  /=47,  x=t^t„  y=i3,  z=i. 
One  would  have  thought  that  after  so  brilliant  a  beginning,  the 
sciences  transplanted  from  Mohammedan  to  Christian  soil  would 
have  enjoyed  a  steady  and  vigorous  development.  But  this  was  not 
the  case.     Durins;  the  fourteenth  and  fifteenth  centuries,  the  mathe- 


THE  FmST  AWAKENING  AND  ITS  SEQUEL         125 

matical  science  was  almost  stationary.  Long  wars  absorbed  the  ener- 
gies of  the  people  and  thereby  kept  back  the  growth  of  the  sciences. 
The  death  of  Frederick  II  in  1254  was  followed  by  a  period  of  con- 
fusion in  Germany.  The  German  emperors  and  the  popes  were  con- 
tinually quarrelling,  and  Italy  was  inevitably  drawn  into  the  struggles 
between  the  Guelphs  and  the  Ghibellines.  France  and  England  were 
engaged  in  the  Hundred  Years'  War  (1338-1453).  Then  followed  in 
England  the  Wars  of  the  Roses.  The  growth  of  science  was  retarded 
not  only  by  war,  but  also  by  the  injurious  influence  of  scholastic  phi- 
losophy. The  intellectual  leaders  of  those  times  quarrelled  over  subtle 
subjects  in  metaphysics  and  theology.  Frivolous  questions,  such  as 
"  How  many  angels  can  stand  on  the  point  of  a  needle?  "  were  discussed 
with  great  interest.  Indistinctness  and  confusion  of  ideas  charac- 
terized the  reasoning  during  this  period.  The  writers  on  mathematics 
during  this  period  were  not  few  in  number,  but  their  scientific  efforts 
were  vitiated  by  the  method  of  scholastic  thinking.  Though  they 
possessed  the  Elements  of  Euclid,  yet  the  true  nature  of  a  mathematical 
proof  was  so  Httle  understood,  that  Hankel  believes  it  no  exaggeration 
to  say  that  "since  Fibonacci,  not  a  single  proof,  not  borrowed  from 
Euclid,  can  be  found  in  the  whole  literature  of  these  ages,  which  fulfils 
all  necessary  conditions." 

The  only  noticeable  advance  is  a  simplification  of  numerical  opera- 
tions and  a  more  extended  application  of  them.  Among  the  ItaUans 
are  evidences  of  an  early  maturity  of  arithmetic.  Peacock  ^  says: 
The  Tuscans  generally,  and  the  Florentines  in  particular,  whose  city 
was  the  cradle  of  the  literature  and  arts  of  the  thirteenth  and  four- 
teenth centuries,  were  celebrated  for  their  knowledge  of  arithmetic 
and  book-keeping,  which  were  so  necessary  for  their  extensive  com- 
merce; the  Italians  were  in  familiar  possession  of  commercial  arith- 
metic long  before  the  other  nations  of  Europe;  to  them  we  are  indebted 
for  the  formal  introduction  into  books  of  arithmetic,  under  distinct 
heads,  of  questions  in  the  single  and  double  rule  of  three,  loss  and  gain, 
fellowship,  exchange,  simple  and  compound  interest,  discount,  and 
so  on. 

There  was  also  a  slow  improvement  in  the  algebraic  notation.  The 
Hindu  algebra  possessed  a  tolerable  symbolic  notation,  which  was, 
however,  completely  ignored  by  the  Mohammedans.  In  this  respect, 
Arabic  algebra  approached  much  more  closely  to  that  of  Diophantus, 
which  can  scarcely  be  said  to  employ  symbols  in  a  systematic  way. 
Leonardo  of  Pisa  possessed  no  algebraic  symbolism.  Like  the  early 
Arabs,  he  expressed  the  relations  of  magnitudes  to  each  other  by  lines 
or  in  words.  But  in  the  mathematical  writings  of  Chuquet  (1484),  of 
Widmann  (1489)  and  of  the  monk  Luca  Pacioli  (also  called  Lucas  de 
Burgo  sepulchri)  symbols  began  to  appear.   PascioU's  consisted  merely 

^  G.  Peaccck,  op.  cil.,  1847,  p.  429. 


126  A  HISTORY  OF  MATHEMATICS 

in  abbreviations  of  Italian  words,  such  as  p  for  piu  (more),  ni  for  mene 
(less),  CO  for  cosa  (the  unknown  x),  ce  for  censo  {x^),  cece  for  censocenso 
{x^),  "Our  present  notation  has  arisen  by  almost  insensible  degrees 
as  convenience  suggested  different  marks  of  abbreviation  to  different 
authors;  and  that  perfect  symbolic  language  which  addresses  itself 
solely  to  the  eye,  and  enables  us  to  take  in  at  a  glance  the  most  com- 
plicated relations  of  quantity,  is  the  result  of  a  large  series  of  small  im- 
provements." ^ 

We  shall  now  mention  a  few  authors  who  lived  during  the  thirteenth 
and  fourteenth  and  the  first  half  of  the  fifteenth  centuries. 

We  begin  with  the  philosophic  writings  of  Thomas  Aquinas  (1225- 
1274),  the  great  ItaUan  philosopher  of  the  Middle  Ages,  who  gave  in 
the  completest  form  the  ideas  of  Origen  on  infinity.  Aquinas'  notion 
of  a  continuum,  particularly  a  linear  continuum,  made  it  potentially 
divisible  to  infinity,  since  practically  the  divisions  could  not  be  carried 
out  to  infinity.  There  was,  therefore,  no  minimum  line.  On  the  other 
hand,  the  point  is  not  a  constituent  part  of  the  hne,  since  it  does  not 
possess  the  property  of  infinite  divisibility  that  parts  of  a  Ime  possess, 
nor  can  the  continuum  be  constructed  out  of  points.  However,  a 
point  by  its  motion  has  the  capacity  of  generating  a  line."  _  This  con- 
tinuum held  a  firm  ascendancy  over  the  ancient  atomistic  doctrine 
which  assumed  matter  to  be  composed  of  very  small,  indivisible  par- 
ticles. No  continuum  superior  to  this  was  created  before  the  nine- 
teenth century.  Aquinas  explains  Zeno's  arguments  against  motion, 
as  they  are  given  by  Aristotle,  but  hardly  presents  any  new  point  of 
view.  The  Englishman,  Roger  Bacon  [i2i4(?)-i294]  likewise  argued 
against  a  continuum  of  indivisible  parts  different  from  points.  Re- 
newing arguments  presented  by  the  Greeks  and  early  Arabs,  he  held 
that  the  doctrine  of  indivisible  parts  of  uniform  size  would  make  the 
diagonal  of  a  square  commensurable  with  a  side.  Likewise,  if  through 
the  ends  of  an  indivisible  arc  of  a  circle  radii  are  drawn,  these  radii 
intercept  an  arc  on  a  concentric  circle  of  smaller  radius;  from  this  it 
would  follow  that  the  inner  circle  is  of  the  same  length  as  the  outer 
circle,  which  is  impossible.  Bacon  argued  against  infinity.  If  time 
were  infinite,  the  absurdity  would  follow  that  the  part  is  equal  to  the 
whole.  Bacon's  views  were  made  known  more  widely  through  Duns 
Scotus  (i 265-1308),  the  theological  and  philosophical  opponent  of 
Thomas  Aquinas.  However,  both  argued  against  the  existence^  of 
indivisible  parts  (points).  Duns  Scotus  wrote  on  Zeno's  paradoxies, 
but  without  reaching  new  points  of  view.  His  commentaries^  were 
annotated  later  by  the  Italian  theologian,  Franciscus  de  Pitigianis, 
who  expressed  himself  in  favor  of  the  admission  of  the  actual  infinity 
to  explain  the  "Dichotomy"  and  the  "Achilles,"  but  fails  to  ade- 
quately elaborate  the  subject.     Scholastic  ideas  on  infinity  and  the 

1  J.  F.  W.  Herschel,  "Mathematics"  in  Edinburgh  Encyclopoedia. 

2  C.  R.  Wallner,  in  Bibllolhcca  malhemalica,  3.  F.,  Bd.  IV,  1903,  pp.  2y,  30. 


THE  FIRST  AWAKENING  AND  ITS  SEQUEL  127 

:ontinuum  find  expression  in  the  writings  of  Bradwardine,  the  Eng- 
lish doctor  profundis} 

About  the  time  of  Leonardo  of  Pisa  (1200  A.  i).),  lived  the  German 
monk  Jordanus  Nemorarius  (?-i237),  who  wrote  a  once  famous  work 
on  the  properties  of  numbers,  printed  in  1496  and  modelled  after  the 
arithmetic  of  Boethius.  The  most  trifling  numeral  properties  are 
treated  with  nauseating  pedantry  and  prolixity.  A  practical  arith- 
metic based  on  the  Hindu  notation  was  also  written  by  him.  John 
Halifax  (Sacro  Bosco,  died  1256)  taught  in  Paris  and  made  an  extract 
from  the  Almagest  containing  only  the  most  elementary  parts  of  that 
work.  This  extract  was  for  nearly  400  years  a  work  of  great  popular- 
ity and  standard  authority,  as  was  also  his  arithmetical  work,  the 
Tractatus  de  arte  numerandi.  Other  prominent  writers  are  Albertus 
Magnus  (ii93?-i28o)  and  Georg  Peurbach  (1423-1461)  in  Ger- 
many. It  appears  that  here  and  there  some  of  our  modern  ideas  were 
anticipated  by  writers  of  the  Middle  Ages.  Thus,  Nicole  Oresme 
Tabout  1323-13S2),  a  bishop  in  Normandy,  first  conceived  the  notion  of 
fractional  powers,  afterwards  rediscovered  by  Stevin,  and  suggested  a 
notation.    Since  4^  =  64,  and  64^  =  8,  Oresme  concluded  that  4^^  =  8. 

In  his  notation,  4^^  is  expressed,  |  ip.||4>  or  I  j^  4.     Some  of  the 

mathematicians  of  the  Middle  Ages  possessed  some  idea  of  a  function. 
Oresme  even  attempted  a  graphic  representation.  But  of  a  numeric 
dependance  of  one  quantity  upon  another,  as  found  in  Descartes, 
there  is  no  trace  among  them.^ 

In  an  unpubhshed  manuscript  Oresme  found  the  sum  of  the  infinite 
series  i+ 1+1+115+  31+  •  •  in  inf.  Such  recurrent  infinite  series  were 
formerly  supposed  to  have  made  their  first  appearance  in  the  eight- 
eenth century.  The  use  of  infinite  series  is  explained  also  in  the  Liber 
de  triplici  motu,  by  the  Portuguese  mathematician  Alvarus  Thomas,^ 
in  1509.  He  gives  the  division  of  a  fine-segment  into  parts  represent- 
ing the  terms  of  a  convergent  geometric  series;  that  is,  a  segment  AB 
is  divided  into  parts  such  that  ^5  :PiB  =  PiB  :P2B=  .  .  =PiB  :Pi+i 
B=  .  .  Such  a  division  of  a  line-segment  occurs  later  in  Napier's 
kinematical  discussion  of  logarithms. 

Thomas  Bradwardine  (about  1 290-1349),  archbishop  of  Canter- 
bury, studied  star-polygons.  The  first  appearance  of  such  polygons  was 
with  Pythagoras  and  his  school.  We  next  meet  with  such  polygons 
in  the  geometry  of  Boethius  and  also  in  the  translation  of  Euclid  from 
the  Arabic  by  Athelard  of  Bath.  To  England  falls  the  honor  of  hav- 
ing produced  the  earliest  European  writers  on  trigonometry.     The 

^  F.  Cajori,  Americ.  Afatlt.  Monlhty,  Vol.  22,  1Q15,  pp.  45-47- 

^H.  Wieleitner  in  BiblioUicca  mathematica,  3.  S.,  Vol.  13,  1913,  pp.  1 15-145. 

'See  Etudes  sur  Leonard  da  Vinci,  Vol.  Ill,  Paris,  1913,  pp.  393,  540,  541,  by 

I'ierrc  Duhem  (1861-1916)  of  the  University  of  Bordeaux;  see  also  Wwtleitner  in 

Bibliothcca  inathcmalica,  Vol.  14,  1914,  pp.  150-168. 


128  A  HISTORY  OF  MATHEMATICS 

writings  of  Bradwardine,  of  Richard  of  Wallingford,  and  John  Maud- 
ith,  both  professors  at  Oxford,  and  of  Simon  Bredon  of  Winchecombe, 
contain  trigonometry  drawn  from  Arabic  sources. 

The  works  of  the  Greek  monk  Maximus  Planudes  (abcmt  1260- 
13 10),  are  of  interest  only  as  showing  that  the  Hindu  numerals  were 
then  known  in  Greece.  A  writer  belonging,  Uke  Planudes,  to  the  By- 
zantine school,  was  Manuel  Moschopulus  who  lived  in  Constantino- 
ple in  the  early  part  of  the  fourteenth  century.  To  him  appears  to  be 
due  the  introduction  into  Europe  of  magic  squares.  He  wrote  a  treatise 
on  this  subject.  Magic  squares  were  known  before  this  to  the  Arabs 
and  Japanese;  they  originated  with  the  Chinese.  Mediaeval  astrol- 
ogers and  physicians  believed  them  to  possess  mystical  properties  and 
to  be  a  charm  against  plague,  when  engraved  on  silver  plate. 

Recently  there  has  been  printed  a  Hebrew  arithmetical  work  by 
the  French  Jew,  Levi  ben  Gerson,  written  in  1321,^  and  handed  down 
in  several  manuscripts.  It  contains  formulas  for  the  number  of  per- 
mutations and  combinations  of  n  things  taken  ;^  at  a  time.  It  is  worthy 
of  note  that  the  earliest  practical  arithmetic  known  to  have  been 
brought  out  in  print  appeared  anonymously  in  Treviso,  Italy,  in  1478, 
and  is  referred  to  as  the  "Treviso  arithmetic."  Four  years  later,  in 
1482,  came  out  at  Bamberg  the  first  printed  German  arithmetic.  It 
is  by  Ulrlch  Wagner,  a  teacher  of  arithmetic  at  Niirnberg.  It  was 
printed  on  parchment,  but  only  fragments  of  one  copy  are  now  extant.^ 

According  to  Enestrom,  Ph.  Calandri's  De  arithmelrica  opusculum, 
Florence,  1491,  is  the  first  printed  treatise  containing  the  word  "zero"; 
it  is  found  in  some  fourteenth  century  manuscripts. 

In  1494  was  printed  the  Summa  de  Arithmetica,  Geometria,  Propor- 
tione  et  ProportionaUta,  written  by  the  Tuscan  monk  Luca  Pacioli 
(1445-15 14?),  who,  as  we  remarked,  introduced  several  symbols  in 
algebra.  This  contains  all  the  knowledge  of  his  day  on  arithmetic, 
algebra,  and  trigonometry,  and  is  the  first  comprehensive  work  which 
appeared  after  the  Liber  Abaci  of  Fibonacci.  It  contains  Uttle  of  im- 
portance which  cannot  be  found  in  Fibonacci's  great  work,  published 
three  centuries  earlier.  Pacioli  came  in  personal  touch  with  two  ar- 
tists who  were  also  mathematicians,  Leonardo  da  Vinci  ^  (145 2-1 5 19) 
and  Pier  delta  Francesca  (1416-1492).  Da  Vinci  inscribed  regular 
polygons  in  circles,  but  did  not  distinguish  between  accurate  and  ap- 
proximate constructions.  It  is  interesting  to  note  that  da  Vinci  was 
familiar  with  the  Greek  text  of  Archimedes  on  the  measurement  of 
the  circle.  Pier  della  Francesca  advanced  the  theory  of  perspective, 
and  left  a  manuscript  on  regular  soHds  which  was  published  by 

1  Bibliotheca  mathematica,  3.  S.,  Vol.  14,  1916,  p.  261. 

2  See  D.  E.  Smith,  Rara  arithmetica,  Boston  and  London,  1908,  pp.  3,  12,  15; 
F.  Unger,  Methodili  der  Praktischen  Arithmetik  in  Historischer  Enlwickelung,  Leip- 
zig, 1888,  p.  39. 

"  Consult  P.  Duhem's  Etudes  sur  Leonard  de  Vinci,  Paris,  1909. 


THE  FIRST  AWAKENING  AND  ITS  SEQUEL         129 

Pacioli  in  1509  as  his  own  work,  in  a  book  entitled,  Divina  pro- 
porlione. 

Perhaps  the  greatest  result  of  the  influx  of  Arabic  learning  was  the 
establishment  of  universities.  What  was  their  attitude  toward  mathe- 
matics? The  University  of  Paris,  so  famous  at  the  beginning  of  the 
twelfth  century  under  the  teachings  of  Abelard  paid  but  little  atten- 
tion to  this  science  during  the  Middle  Ages.  Geometry  was  neglected, 
and  Aristotle's  logic  was  the  favorite  study.  In  1336,  a  rule  was  in- 
troduced that  no  student  should  take  a  degree  without  attending  lec- 
tures on  mathematics,  and  from  a  commentary  on  the  first  six  books 
of  Euclid,  dated  1536,  it  appears  that  candidates  for  the  degree  of 
A.  M.  had  to  give  an  oath  that  they  had  attended  lectures  on  these 
books.^  Examinations,  when  held  at  all,  probably  did  not  extend  be- 
yond the  first  book,  as  is  shown  by  the  nickname  "magister  mathe- 
seos,"  applied  to  the  Theorem  of  Pythagoras,  the  last  in  the  first  book. 
More  attention  was  paid  to  mathematics  at  the  University  of  Prague, 
founded  1384.  For  the  Baccalaureate  degree,  students  were  required 
to  take  lectures  on  Sacro  Bosco's  famous  work  on  astronomy.  Of  can- 
didates for  the  A.M.  were  required  not  only  the  six  books  of  Euclid, 
but  an  additional  knowledge  of  applied  mathematics.  Lectures  were 
given  on  the  Almagest.  At  the  University  of  Leipzig,  the  daughter  of 
Prague,  and  at  Cologne,  less  work  was  required,  and,  as  late  as  the 
sixteenth  century,  the  same  requirements  were  made  at  these  as  at 
Prague  in  the  fourteenth.  The  universities  of  Bologna,  Padua,  Pisa, 
occupied  similar  positions  to  the  ones  in  Germany,  only  that  purely 
astrological  lectures  were  given  in  place  of  lectures  on  the  Almagest. 
At  Oxford,  in  the  middle  of  the  fifteenth  century,  the  first  two  books 
of  Euclid  were  read.^ 

Thus  it  will  be  seen  that  the  study  of  mathematics  was  maintained 
at  the  universities  only  in  a  half-hearted  manner.  No  great  mathe- 
matician and  teacher  appeared,  to  inspire  the  students.  The  best 
energies  of  the  schoolmen  were  expended  upon  the  stupid  subtleties  of 
their  philosophy.  The  genius  of  Leonardo  of  Pisa  left  no  permanent 
impress  upon  the  age,  and  another  Renaissance  of  mathematics  was 
wanted. 

1  H.  Hankel,  op.  cit.,  p.  355.  'J.  Gow,  op.  cit.,  p.  207. 


EUROPE  DURING  THE  SIXTEENTH,  SEVENTEENTH 
AND  EIGHTEENTH  CENTURIES 

We  find  it  convenient  to  choose  the  time  of  the  capture  of  Constan- 
tinople by  the  Turks  as  the  date  at  which  the  Middle  Ages  ended  and 
Modern  Times  began.  In  1453,  the  Turks  battered  the  walls  of  this 
celebrated  metropolis  with  cannon,  and  finally  captured  the  city;  the 
Byzantine  Empire  fell,  to  rise  no  more.  Calamitous  as  was  this  event 
to  the  East,  it  acted  favorably  upon  the  progress  of  learning  in  the 
West.  A  great  number  of  learned  Greeks  fled  into  Italy,  bringing  with 
them  precious  manuscripts  of  Greek  literature.  This  contributed 
vastly  to  the  reviving  of  classic  learning.  Up  to  this  time,  Greek  mas- 
ters were  known  only  through  the  often  very  corrupt  Arabic  manu- 
scripts, but  now  they  began  to  be  studied  from  original  sources  and 
in  their  own  language.  The  first  English  translation  of  Euclid  was 
made  in  1570  from  the  Greek  by  Sir  Henry  Billingsley,  assisted  by 
John  Dee}  About  the  middle  of  the  fifteenth  century,  printing  was 
invented;  books  became  cheap  and  plentiful;  the  printing-press  trans- 
formed Europe  into  an  audience-room.  Near  the  close  of  the  fifteenth 
century,  America  was  discovered,  and,  soon  after,  the  earth  was  cir- 
cumnavigated. The  pulse  and  pace  of  the  world  began  to  quicken. 
Men's  minds  became  less  servile;  they  became  clearer  and  stronger. 
The  indistinctness  of  thought,  which  was  the  characteristic  feature  of 
mediceval  learning,  began  to  be  remedied  chiefly  by  the  steady  cultiva- 
tion of  Pure  Mathematics  and  Astronomy.  Dogmatism  was  attacked; 
there  arose  a  long  struggle  with  the  authority  of  the  Church  and  the 
established  schools  of  philosophy.  The  Copernican  System  was  set 
up  in  opposition  to  the  time-honored  Ptolemaic  System.  The  long 
and  eager  contest  between  the  two  culminated  in  a  crisis  at  the  time 
of  Galileo,  and  resulted  in  the  victory  of  the  new  system.  Thus,  by 
slow  degrees,  the  minds  of  men  were  cut  adrift  from  their  old  scholastic 
moorings  and  sent  forth  on  the  wide  sea  of  scientific  inquiry,  to  dis- 
cover new  islands  and  continents  of  truth. 

The  Renaissance 

With  the  sixteenth  century  began  a  period  of  increased  intellectual 
activity.  The  human  mind  made  a  vast  effort  to  achieve  its  freedom. 
Attempts  at  its  emancipation  from  Church  authority  had  been  made 
before,  but  they  were  stifled  and  rendered  abortive.  The  first  great 
and  successful  revolt  against  ecclesiastical  authority  was  made  in 
'  G.  B.  Halsted  in  Am.  Jour,  of  Math.,  Vol.  II,  1879. 
130 


THE  RENAISSANCE  i,:?i 

Germany.  The  new  desire  for  judging  freely  and  independently  in 
matters  of  religion  was  preceded  and  accompanied  by  a  growing  spirit 
of  scientific  inquiry.  Thus  it  was  that,  for  a  time,  Germany  led  the 
van  in  science.  She  produced  Regiomonlaniis,  Copernicus,  R/kbHcus 
and  Kepler,  at  a  period  when  France  and  England  had,  as  yet,  brought 
forth  hardly  any  great  scientific  thinkers.  This  remarkable  scientific 
productiveness  was  no  doubt  due,  to  a  great  extent,  to  the  commercial 
prosperity  of  Germany.  Material  prosperity  is  an  essential  condition 
for  the  progress  of  knowledge.  As  long  as  every  individual  is  obliged 
to  collect  the  necessaries  for  his  subsistence,  there  can  be  no  leisure 
for  higher  pursuits.  At  this  time,  Germany  had  accumulated  con- 
siderable wealth.  The  Hanseatic  League  commanded  the  trade  of 
the  North.  Close  commercial  relations  existed  between  Germany  and 
Italy.  Italy,  too,  excelled  in  commercial  activity  and  enterprise. 
We  need  only  mention  Venice,  whose  glory  began  with  the  crusades, 
and  Florence,  with  her  bankers  and  her  manufacturers  of  silk  and  wool. 
These  two  cities  became  great  intellectual  centres.  Thus,  Italy,  too, 
produced  men  in  art,  literature,  and  science,  who  shone  forth  in  fullest 
splendor.  In  fact,  Italy  was  the  fatherland  of  what  is  termed  the  Re- 
naissance. 

For  the  first  great  contributions  to  the  mathematical  sciences  we 
must,  therefore,  look  to  Italy  and  Germany.  In  Italy  brilliant  acces- 
sions were  made  to  algebra,  in  Germany  progress  was  made  in  astron- 
omy and  trigonometry. 

On  the  threshold  of  this  new  era  we  meet  in  Germany  with  the  figure 
of  John  IMueller,  more  generally  called  Regiomontanus  (1436- 1476). 
Chiefly  to  him  we  owe  the  revival  of  trigonometry.  He  studied  as- 
tronomy and  trigonometry  at  Vienna  under  the  celebrated  George 
Peurbach.  The  latter  perceived  that  the  existing  Latin  translations 
of  the  Almagest  were  full  of  errors,  and  that  Arabic  authors  had  not 
remained  true  to  the  Greek  original.  Peurbach  therefore  began  to 
make  a  translation  directly  from  the  Greek.  But  he  did  not  live  to 
finish  it.  His  work  was  continued  by  Regiomontanus,  who  went  be- 
yond his  master.  Regiomontanus  learned  the  Greek  language  from 
Cardinal  Bessarion,  whom  he  followed  to  Italy,  where  he  remained 
eight  years  collecting  manuscripts  from  Greeks  who  had  fled  thither 
from  the  Turks.  In  addition  to  the  translation  of  and  the  commen- 
tary on  the  Almagest,  he  prepared  translations  of  the  Conies  of  Apol- 
lonius,  of  Archimedes,  and  of  the  mechanical  works  of  Heron.  Regio- 
montanus and  Peurbach  adopted  the  Hindu  sine  in  place  of  the  Greek 
chord  of  double  the  arc.  The  Greeks  and  afterwards  the  Arabs  divided 
the  radius  into  60  equal  parts,  and  each  of  these  again  into  60  smaller 
ones.  The  Hindu  expressed  the  length  of  the  radius  by  parts  of  the 
circumference,  saying  that  of  the  21,600  equal  divisions  of  the  latter, 
it  took  3438  to  measure  the  radius.  Regiomontanus,  to  secure  greater 
precision,    constructed  one  table  of  sines  on  a  radius  divided  into 


132  A  HISTORY  OF  MATHEMATICS 

6oc,ooo  parts,  and  another  on  a  radius  divided  decimally  into 
10,000,000  divisions.  He  emphasized  the  use  of  the  tangent  in  trigo- 
nometry. Following  out  some  ideas  of  his  master,  he  calculated  a 
table  of  tangents.  German  mathematicians  were  not  the  first  Euro- 
peans to  use  this  function.  In  England  it  was  known  a  century  earlier 
to  Bradwardine,  who  speaks  of  tangent  (umbra  versa)  and  cotangent 
{umbra  recta),  and  to  John  Maudith.  Even  earlier,  in  the  twelfth 
century,  the  umbra  versa  and  umbra  recta  are  used  in  a  translation  from 
Arabic  into  Latin,  effected  by  Gerard  of  Cremona,  of  the  Toledian 
Tables  ol  /il-Zarkali,  who  lived  in  Toledo  about  loSo.  Regiomontanus 
was  the  author  of  an  arithmetic  and  also  of  a  complete  treatise  on 
trigonometry,  containing  solutions  of  both  plane  and  spherical  tri- 
angles. Some  innovations  in  trigonometry,  formerly  attributed  to 
Regiomontanus,  are  now  known  to  have  been  introduced  by  the  Arabs 
before  him.  Nevertheless,  much  credit  is  due  to  him.  His  complete 
mastery  of  astronomy  and  mathematics,  and  his  enthusiasm  for  them, 
were  of  far-reaching  influence  throughout  Germany.  So  great  was  his 
reputation,  that  Pope  Sixtus  IV  called  him  to  Italy  to  improve  the 
calendar.  Regiomontanus  left  his  beloved  city  of  Nurnberg  for  Rome, 
where  he  died  in  the  following  year. 

After  the  time  of  Peurbach  and  Regiomontanus,  trigonometry  and 
especially  the  calculation  of  tables  continued  to  occupy  German  schol- 
ars. More  refined  astronomical  instruments  were  made,  which  gave 
observations  of  greater  precision;  but  these  would  have  been  useless 
without  trigonometrical  tables  of  corresponding  accuracy.  Of  the  sev- 
eral tables  calculated,  that  by  Georg  Joachim  of  Feldkirch  in  Tyrol,  gen- 
erally called  Rhaeticus  (1514-1567)  deserves  special  mention.  He  cal- 
culated a  table  of  sines  with  the  radius  =10,000,000,000  and  from  10" 
to  10";  and,  later  on,  another  with  the  radius  =1,000,000,000,000,000, 
and  proceeding  from  10"  to  10".  He  began  also  the  construction  of 
tables  of  tangents  and  secants,  to  be  carried  to  the  same  degree  of 
accuracy;  but  he  died  before  finishing  them.  For  twelve  years  he  had 
had  in  continual  employment  several  calculators.  The  work  was  com- 
pleted in  1596  by  his  pupil,  Valentine  Otho  (i55o?-i6o5).  This  was 
indeed  a  gigantic  work, — a  monument  of  German  diligence  and  inde- 
fatigable perseverance.  The  tables  were  repubhshed  in  1613  by  Bar- 
tholomaus  Pitiscus  (1561-1613)  of  Heidelberg,  who  spared  no  pains 
to  free  them  of  errors.  Pitiscus  was  perhaps  the  first  to  use  the  word 
"trigonometry."  Astronomical  tables  of  so  great  a  degree  of  accu- 
racy had  never  been  dreamed  of  by  the  Greeks,  Hindus,  or  Arabs. 
That  Rhajticus  was  not  a  ready  calculator  only,  is  indicated  by  his 
views  on  trigonometrical  lines.  Up  to  his  time,  the  trigonometric 
functions  had  been  considered  always  with  relation  to  the  arc;  he  was 
the  first  to  construct  the  right  triangle  and  to  make  them  depend  di- 
rectly upon  its  angles.  It  was  from  the  right  triangle  that  Rhaeticus 
got  his  idea  of  calculating  the  h)^otenuse;  i.  e.  he  was  the  first  to  plan 


THE  RENAISSANCE 


13,^ 


a  table  of  secants.    Good  work  in  trigonometry  was  done  also  by  Vieta 
and  Romanus. 

We  shall  now  leave  the  subject  of  trigonometry  to  witness  the  prog- 
ress in  the  solution  of  algebraical  equations.  To  do  so,  we  must  quit 
Germany  for  Italy.  The  first  comprehensive  algebra  printed  was  that 
of  Luca  Pacioli.  He  closes  his  book  by  saying  that  the  solution  of  the 
equations  x^-{-mx=n,  x^-i-n=mx  is  as  impossible  at  the  present  state 
of  science  as  the  quadrature  of  the  circle.  This  remark  doubtless  stim- 
ulated thought.  The  first  step  in  the  algebraic  solution  of  cubics  was 
taken  by  Scipione  del  Ferro  (1465-1526),  a  professor  of  mathematics 
at  Bologna,  who  solved  the  equation  x^-\-!nx=ii.  He  imparted  it  to 
his  pupil,  Floridas,  in  1505,  but  did  not  publish  it.  It  was  the  practice 
in  those  days  and  for  two  centuries  afterwards  to  keep  discoveries 
secret,  in  order  to  secure  by  that  means  an  advantage  over  rivals  by 
proposing  problems  beyond  their  reach.  This  practice  gave  rise  to 
numberless  disputes  regarding  the  priority  of  inventions.  A  second 
solution  of  cubics  was  given  by  Nicolo  of  Brescia  [i499(?)-i557]. 
WTien  a  boy  of  six,  Nicolo  was  so  badly  cut  by  a  French  soldier  that 
he  never  again  gained  the  free  use  of  his  tongue.  Hence  he  was  called 
Tartaglia,  /.  e.  the  stammerer.  His  widowed  mother  being  too  poor  to 
pay  his  tuition  in  school,  he  learned  to  read  and  picked  up  a  knowledge 
of  Latin,  Greek,  and  mathematics  by  himself.  Possessing  a  mind  of 
extraordinary  power,  he  was  able  to  appear  as  teacher  of  mathematics 
at  an  early  age.  He  taught  in  Venice,  then  in  Brescia,  and  later  again 
in  Venice.  In  1530,  one  Colla  proposed  him  several  problems,  one 
leading  to  the  equation  x^-\-px-  =  q.  Tartaglia  found  an  imperfect 
method  for  solving  this,  but  kept  it  secret.  He  spoke  about  his  secret 
in  public  and  thus  caused  Del  Ferro's  pupil,  Floridas,  to  proclaim  his 
own  knowledge  of  the  form  x^-\-mx=n.  Tartaglia,  believing  him  to 
be  a  mediocrist  and  braggart,  challenged  him  to  a  public  discussion,  to 
take  place  on  the  22d  of  February,  1535.  Hearing,  meanwhile,  that 
his  rival  had  gotten  the  method  from  a  deceased  master,  and  fearing 
that  he  would  be  beaten  in  the  contest,  Tartaglia  put  in  all  the  zeal, 
industry,  and  skill  to  find  the  rule  for  the  equations,  and  he  succeeded 
in  it  ten  days  before  the  appointed  date,  as  he  himself  modestly  says.^ 
The  most  difficult  step  was,  no  doubt,  the  passing  from  quadratic  ir- 
rationals, used  in  operating  from  time  of  old,  to  cubic  irrationals. 
Placing  x=\/t—-\/ii,  Tartaglia  perceived  that  the  irrationals  dis- 
appeared from  the  equation  x^  =  mx  —  n,  making  n  =  t—u.  But  this 
last  equality,  together  with  {\my  —  tu,  gives  at  once 


.|(")+(j)+;-  -^l©■+(?)'-I• 


This  is  Tartaglia's  solution  of  x^-\-mx  =  n.    On  the  13th  of  February, 

he  found  a  similar  solution  for  x^  =  mx-\-n.    The  contest  began  on  the 

^  H.  Hankel,  op.  cit.,  p.  362. 


134  A  HISTORY  OF  MATHEMATICS 

2 2d.  Each  contestant  proposed  thirty  problems.  The  one  who  could 
solve  the  greatest  number  within  fifty  days  should  be  the  victor.  Tar- 
taglia  solved  the  thirty  problems  proposed  by  Floridas  in  two  hours; 
Floridas  could  not  solve  any  of  Tartaglia's.  From  now  on,  Tartaglia 
studied  cubic  equations  with  a  will.  In  1541  he  discovered  a  general 
solution  for  the  cubic  .\;^=±=  px^  ==^q,  by  transforming  it  into  the  form 
x^d=mx=^jt.  The  news  of  Tartaglia's  victory  spread  all  over  Italy. 
Tartaglia  was  entreated  to  make  known  his  method,  but  he  declined 
to  do  so,  saying  that  after  his  completion  of  the  translation  from  the 
Greek  of  Euclid  and  Archimedes,  he  would  publish  a  large  algebra 
containing  his  method.  But  a  scholar  from  Milan,  named  Hieronimo 
Cardano  (1501-1576),  after  many  solicitations,  and  after  giving  the 
most  solemn  and  sacred  promises  of  secrecy,  succeeded  in  obtaining 
from  Tartaglia  a  knowledge  of  his  rules.  Cardan  was  a  singular  mix- 
ture of  genius,  folly,  self-conceit  and  mysticism.  He  was  successively 
professor  of  mathematics  and  medicine  at  Milan,  Pavia  and  Bologna, 
In  1570  he  was  imprisoned  for  debt.  Later  he  went  to  Rome,  v/as 
admitted  to  the  college  of  physicians  and  was  pensioned  by  the  pope. 
At  this  time  Cardan  was  writing  his  Ars  Magna,  and  he  knew  no 
better  way  to  crown  his  work  than  by  inserting  the  much  sought  for 
rules  for  solving  cubics.  Thus  Cardan  broke  his  most  solemn  vows, 
and  published  in  1545  in  his  Ars  Magna  Tartaglia's  solution  of  cubics. 
However,  Cardan  did  credit  "his  friend  Tartaglia"  with  the  discovery 
of  the  rule.  Nevertheless,  Tartaglia  became  desperate.  His  most 
cherished  hope,  of  giving  to  the  world  an  immortal  work  which  should 
be  the  monument  of  his  deep  learning  and  power  for  original  research, 
was  suddenly  destroyed;  for  the  crown  intended  for  his  work  had 
been  snatched  away.  His  first  step  was  to  write  a  history  of  his  in- 
vention; but,  to  completely  annihilate  his  enemies,  he  challenged 
Cardan  and  his  pupil  Lodovico  Ferrari  to  a  contest:  each  party 
should  propose  thirty-one  questions  to  be  solved  by  the  other  within 
fifteen  days.  Tartaglia  solved  most  questions  in  seven  days,  but  the 
other  party  did  not  send  in  their  solutions  before  the  expiration  of  the 
fifth  month;  moreover,  all  their  solutions  except  one  were  wrong.  A 
replication  and  a  rejoinder  followed.  Endless  were  the  problems  pro- 
posed and  solved  on  both  sides.  The  dispute  produced  much  chagrin 
and  heart-burnings  to  the  parties,  and  to  Tartaglia  especially,  who 
met  with  many  other  disappointments.  After  having  recovered  him- 
self again,  Tartaglia  began,  in  1556,  the  publication  of  the  work  which 
he  had  had  in  his  mind  for  so  long;  but  he  died  before  he  reached  the 
consideration  of  cubic  equations.  Thus  the  fondest  wish  of  his  life  re- 
mained unfulfilled.  How  much  credit  for  the  algebraic  solution  of  the 
general  cubic  is  due  to  Tartaglia  and  how  much  to  Del  Ferro  it  is  now 
impossible  to  ascertain  definitely,  Del  Ferro's  researches  were  never 
published  and  were  lost.  We  know  of  them  only  through  the  remarks 
of  Cardan  and  his  pupil  L.  Ferrari  who  say  that  Del  P>rro's  and  Tar 


THE  RENAISSANCE  135 

taglia's  methods  were  alike.  Certain  it  is  that  the  customary  desig- 
nation, "Cardan's  solution  of  the  cubic"  ascribes  to  Cardan  what 
belongs  to  one  or  the  other  of  his  predecessors. 

Remarkable  is  the  great  interest  that  the  solution  of  cubics  excited 
throughout  Italy.  It  is  but  natural  that  after  this  great  conquest 
mathematicians  should  attack  bi-quadratic  equations.  As  in  the  case 
of  cubics,  so  here,  the  first  impulse  was  given  by  CoUa,  who,  in  1540, 
proposed  for  solution  the  equation  .v*'+6.v-+36  =  6o.r.  To  be  sure, 
Cardan  had  studied  particular  cases  as  early  as  1539.  Thus  he  solved 
the  equation  i3.v-  =  .v^+2.v^+2.r+i  by  a  process  similar  to  that  em- 
ployed by  Diophantus  and  the  Hindus;  namely,  by  adding  to  both 
sides  3.x*-  and  thereby  rendering  both  numbers  complete  squares.  But 
Cardan  failed  to  find  a  general  solution;  it  remained  for  his  pupil 
Lodovico  Ferrari  (1522-1565)  of  Bologna  to  make  the  brilliant  dis- 
covery of  the  general  solution  of  bi-quadratic  equations.  Ferrari  re- 
duced Colla's  equation  to  the  form  {x--\-6)-=6ox-\-6x".  In  order  to 
give  also  the  right  member  the  form  of  a  complete  square  he  added  to 
both  members  the  expression  2(x--\-6)y-\-y'^,  containing  a  new  un- 
known quantity  y.  This  gave  him  (.x^+6-l-y)^=  (6+2y).v-+6ox-h 
(I2y+y'").  The  condition  that  the  right  member  be  a  complete  square 
is  expressed  by  the  cubic  equation  (2y+6)  (i2y-|-y-)  =  9oo.  Extract- 
ing the  square  root  of  the  bi-quadratic,  he  got  .v-+6+y=.rV2y+6 

H —  '         .     Solving  the  cubic  for  y  and  substituting,  it  remained 
V2y+6 

only  to  determine  x  from  the  resulting  quadratic.  L.  Ferrari  pursued 
a  similar  method  with  other  numerical  bi-quadratic  equations.^  Car- 
dan had  the  pleasure  of  publishing  this  discovery  in  his  Ars  Magna 
in  1545.  Ferrari's  solution  is  sometimes  ascribed  to  R.  Bombelli,  but 
he  is  no  more  the  discoverer  of  it  than  Cardan  is  of  the  solution  called 
by  his  name. 

To  Cardan  algebra  is  much  indebted.  In  his  Ars  Magna  he  takes 
notice  of  negative  roots  of  an  equation,  caUing  them  fictitious,  while 
the  positive  roots  are  called  real.  He  paid  some  attention  to  compu- 
tations involving  the  square  root  of  negative  numbers,  but  failed 
to  recognize  imaginary  roots.  Cardan  also  observed  the  difficulty 
in  the  irreducible  case  in  the  cubics,  which,  like  the  quadrature  of  the 
circle,  has  since  "so  much  tormented  the  perverse  ingenuity  of  mathe- 
maticians." But  he  did  not  understand  its  nature.  It  remained  for 
Raphael  Bombelli  of  Bologna,  who  published  in  1572  an  algebra  of 
great  merit,  to  point  out  the  reality  of  the  apparently  imaginary  ex- 
pression which  a  root  assumes,  also  to  assign  its  value,  when  rational, 
and  thus  to  lay  the  foundation  of  a  more  intimate  knowledge  of  imagi- 
nary quantities.  Cardan  was  an  inveterate  gambler.  In  1663  there 
was  published  posthumously  his  gambler's  manual,  De  ludo  alecB 
'  H.  Hankel,  oi>.  cit..  d.  i6S. 


136  A  HISTORY  OF  MATHEMATICS 

which  contains  discussions  relating  to  the  chances  favorable  for  throw- 
ing a  particular  number  with  two  dice  and  also  with  three  dice.  Car- 
dan considered  another  problem  in  probabilities.  Stated  in  general 
terms,  the  problem  is:  What  is  the  proper  division  of  a  stake  between 
two  players,  if  the  game  is  interrupted  and  one  player  has  taken  si 
points,  the  other  52  points,  s  points  being  required  to  win.^  Cardan 
gives  the  ratio  (1+2+  .  .  +[5-52])/(i+2+  .  .  +b-5i]),  Tartaglia 
gives  (5+5i-52)/(5+52— 5i).  Both  of  these  answers  are  wrong.  Car- 
dan considered  also  what  later  became  known  as  the  "Petersburg 
problem." 

After  the  brilliant  success  in  solving  equations  of  the  third  and 
fourth  degrees,  there  was  probably  no  one  who  doubted,  that  with 
aid  of  irrationals  of  higher  degrees,  the  solution  of  equations  of  any 
degree  whatever  could  be  found.  But  all  attempts  at  the  algebraic 
solution  of  the  quintic  were  fruitless,  and,  finally,  Abel  demonstrated 
that  all  hopes  of  finding  algebraic  solutions  to  equations  of  higher 
than  the  fourth  degree  were  purely  Utopian. 

Since  no  solution  by  radicals  of  equations  of  higher  degrees  could 
be  found,  there  remained  nothing  else  to  be  done  than  the  devising  of 
processes  by  which  the  real  roots  of  numerical  equations  could  be 
found  by  approximation.  The  Chinese  method  used  by  them  as  early 
as  the  thirteenth  century  was  unknown  in  the  Occident.  We  have 
seen  that  in  the  early  part  of  the  thirteenth  century  Leonardo  of  Pisa 
solved  a  cubic  to  a  high  degree  of  approximation,  but  we  are  ignorant 
of  his  method.  The  earliest  known  process  in  the  Occident  of  ap- 
proaching to  a  root  of  an  affected  numerical  equation  was  invented  by 
Nicolas  Chuquet,  who,  in  1484  at  Lyons,  wrote  a  work  of  high  rank, 
entitled  Le  triparty  en  la  science  des  nomhres.    It  was  not  printed  until 

18S0.-    If  ->.v<-:,  then  Chuquet  takes  the  intermediate  value  —r— 
c         d  c-\-a 

as  a  closer  approximation  to  the  root  x.  He  finds  a  series  of  successive 
intermediate  values.  We  stated  earlier  that  in  1498  the  Arabic  writer 
Miram  Chelebi  gave  a  method  of  solving  x^+(2  =  i':»  which  he  attrib- 
utes to  Atabeddin  Jamshid.  This  cubic  arose  in  the  computation  of 
x=sin  1°. 

The  earliest  printed  method  of  approximation  to  the  roots  of  af- 
fected equations  is  that  of  Cardan,  who  gave  it  in  the  Ars  Magna, 
1545,  under  the  title  of  regula  aurea.  It  is  a  skilful  appUcation  of 
the  rule  of  "false  position,"  and  is  applicable  to  equations  of  any  de- 
gree. This  mode  of  approximation  was  exceedingly  rough,  yet  this 
fact  hardly  explains  why  Clavius,  Stevin  and  Vieta  did  not  refer  to  it. 

1  M.  Cantor,  IT,  2  Aufl.,  1900,  pp.  501,  520,  537. 

2  Printed  in  the  Biilldino  Boncompagiii,  T  xiii,  1880;  see  pp.  653-654.  See  also 
F.  Cajori,  "A  History  of  the  Arithmetical  Methods  of  Approximation  to  the 
Roots  of  Numerical  Equations  of  one  Unknown  Quantity"  in  Colorado  College 
Publication,  General  Series  Nos.  51  and  52,  1910. 


THE  RENAISSANCE  137 

Processes  of  approximation  were  given  by  the  Frenchman  J.  Peletier 
(1554),  the  Italian  R.  Bombelli  (1572),  the  German  R.  Ursus  (i6oi), 
the  Swiss  Joost  Biirgi,  the  German  Pitiscus  (1612),  and  the  Belgian 
Simon  Stevin.  But  far  more  important  than  the  processes  of  these 
men  was  that  of  the  Frenchman,  Francis  Vieta  (1540-1603),  which 
initiates  a  new  era.  It  is  contained  in  a  work  published  at  Paris  in 
1600  by  Marino  Ghetaldi  as  editor,  with  Vieta's  consent,  under  the 
title:  De  numcrosa  protcstatum  purarunt  atque  adfcdarum  ad  exegesin 
resoliUione  tractatus.  His  method  is  not  of  the  nature  of  tlie  rule  of 
"double  false  position,"  used  by  Cardan  and  Biargi,  but  resembles 
the  method  of  ordinary  root-extraction.  Taking  f{x)  =  k,  where  k  is 
taken  positive,  Vieta  separates  the  required  root  from  the  rest,  then 
substitutes  an  approximate  value  for  it  and  shows  that  another  figure 
of  the  root  can  be  obtained  by  division.  A  repetition  of  this  process 
gives  the  next  figure,  and  so  on.  Thus,  in  ::v;^—5r'+5oox=  7905504, 
he  takes  r=20,  then  computes  7905504 -r^+5r^-5oor  and  divides 
the  result  by  a  value  which  in  our  modern  notation  takes  the  form 
\(f{r+s^)  -/(V))l  -5i",  where  n  is  the  degree  of  the  ecjuation  and  ^i  is 
a  unit  of  the  denomination  of  the  digit  next  to  be  found.  Thus,  if  the 
required  root  is  243,  and  r  has  been  taken  to  be  200,  then  5i  is  10;  but 
if  r  is  taken  as  240,  then  5i  is  i.  In  our  example,  where  r=2o,  the 
divisor  is  878295,  and  the  quotient  yields  the  next  digit  of  the  root 
equal  to  4.  We  obtain  a;=2o4-4=24,  the  required  root.  Vieta's 
procedure  was  greatly  admired  by  his  contemporaries,  particularly 
the  Englishmen,  T.  Harriot,  W.  Oughtred  and  J.  Wallis,  each  of  whom 
introduced  some  minor  improvements. 

We  pause  a  moment  to  sketch  the  life  of  Vieta,  the  most  eminent 
French  mathematician  of  the  sixteenth  century.  He  was  born  in 
Poitou  and  died  at  Paris.  He  was  employed  throughout  life  in  the 
service  of  the  state,  under  Henry  III.  and  Henry  IV.  He  was,  there- 
fore, not  a  mathematician  by  profession,  but  his  love  for  the  science 
was  so  great  that  he  remained  in  his  chamber  studying,  sometimes 
several  days  in  succession,  without  eating  and  sleeping  more  than  was 
necessary  to  sustain  himself.  So  great  devotion  to  abstract  science 
is  the  more  remarkable,  because  he  lived  at  a  time  of  incessant  po- 
litical and  religious  turmoil.  During  the  war  against  Spain,  Vieta 
rendered  service  to  Henry  IV  by  deciphering  intercepted  letters  writ- 
ten in  a  species  of  cipher,  and  addressed  by  the  Spanish  Court  to  their 
governor  of  Netherlands.  The  Spaniards  attributed  the  discovery  of 
the  key  to  magic. 

In  1579  Vieta  published  his  Canon  mathemalicus  sen  ad  triangula 
cum  appendicibus,  which  contains  very  remarkable  contributions  to 
trigonometry.  It  gives  the  first  systematic  elaboration  in  the  Occi- 
dent of  the  methods  of  computing  plane  and  spherical  triangles  by 
the  aid  of  the  six  trigonometric  functions.^  He  paid  special  attention 
^  A.  V.  Braunmiihl,  Geschichtc  d:r  Trigonomdry,  I,  Leipzig,  igcx),  p.  i6o. 


13S  A  HISTORY  OF  MATHEMATICS 

also  to  goniometry,  developing  such  relations  as  sin  a  =  sin  (60°+  a) 

—  sin  (60°— a),  CSC a-\-dn a  =  ctn—,    —  c/;ja+c5ca=  tan— ,  with  the 

aid  of  which  he  could  compute  from  the  functions  of  anglee  below 
30°  or  45°,  the  functions  of  the  remaining  angles  below  90°,  essentially 
by  addition  and  subtraction  alone.  Vieta  is  the  first  to  apply  alge- 
braic transformation  to  trigonometry,  particularly  to  the  multisection 
of  angles.  Letting  2  cosa=.v,  he  expresses  cos  na  as  a  function  of  x 
for  all  integers  w<ii;  letting  2  sina=x  and  2  sin  2a=y,  he  expresses 
2.t"~-sin  na  in  terms  of  x  and  y.  Vieta  exclaims:  "Thus  the  analysis 
of  angular  sections  involves  geometric  and  arithmetic  secrets  which 
hitherto  have  been  penetrated  by  no  one." 

An  ambassador  from  Netherlands  once  told  Henry  IV  that  France 
did  not  possess  a  single  geometer  capable  of  solving  a  problem  pro- 
pounded to  geometers  by  a  Belgian  mathematician,  Adrianus  Ro- 
manus.    It  was  the  solution  of  the  equation  of  the  forty- fifth  degree: — 

453'-3795/+95634>'^-  •  •  •  +945/^-45/^+^^=^. 
Henry  IV  called  Vieta,  who,  having  already  pursued  similar  investi- 
gations, saw  at  once  that  this  awe-inspiring  problem  was  simply  the 
equation  by  which  C=2  sin  (f)  was  expressed  in  terms  of  y=2  sin^^  (f>; 
that,  since  45  =  3.3.5,  it  was  necessary  only  to  divide  an  angle  once 
into  5  equal  parts,  and  then  twice  into  3, — a  division  which  could  be 
effected  by  corresponding  equations  of  the  fifth  and  third  degrees. 
Brilliant  was  the  discovery  by  Vieta  of  23  roots  to  this  equation,  in- 
stead of  only  one.  The  reason  why  he  did  not  find  45  solutions,  is 
that  the  remaining  ones  involve  negative  sines,  which  were  unintel- 
ligible to  him.  Detailed  investigations  on  the  famous  old  problem 
of  the  section  of  an  angle  into  an  odd  number  of  equal  parts,  led  Vieta 
to  the  discovery  of  a  trigonometrical  solution  of  Cardan's  irreducible 

case  in  cubics.    He  applied  the  equation  (2  cos  §  0)^— 3(  2  cos-^j  = 

2  COS0  to  the  solution  of  x^-  ^a'^x=a%,  when  a>^b,  by  placing  x= 
2a  cos^cf),  and  determining  0from  b  =  2a  cos0. 

The  main  principle  employed  by  him  in  the  solution  of  equations 
is  that  of  reduction.  He  solves  the  quadratic  by  making  a  suitable 
substitution  which  will  remove  the  term  containing  x  to  the  first  de- 
gree. Like  Cardan,  he  reduces  the  general  expression  of  the  cubic  to 
the  form  x^+nix+n=o;  then,  assuming  x={la—z^)-T-z  and  substi- 
tuting, he  gets  z^—bz^--^^a^  =  o.  Putting  z^=y,  he  has  a  quadratic. 
In  the  solution  of  bi-quadratics,  Vieta  still  remains  true  to  his  principle 
of  reduction.  This  gives  him  the  well-known  cubic  resolvent.  He 
thus  adheres  throughout  to  his  favorite  principle,  and  thereby  in- 
troduces into  algebra  a  uniformity  of  method  which  claims  our  lively 
admiration.  In  Vieta's  algebra  we  discover  a  partial  knowledge  of 
the  relations  existing  between  the  coefficients  and  the  roots  of  an  equa- 


THE  RENAISSANCE 


3q 


tion.  He  shows  that  if  the  coefficient  of  the  second  term  in  an  equa- 
tion of  the  second  degree  is  minus  the  sum  of  two  numbers  whose 
product  is  the  third  term,  then  the  two  numbers  are  roots  of  the  equa- 
tion. Vieta  rejected  all  except  positive  roots;  hence  it  was  impossible 
for  him  to  fully  perceive  the  relations  in  question. 

The  most  epoch-making  innovation  in  algebra  due  to  Vieta  is  the 
denoting  of  general  or  indefinite  quantities  by  letters  of  the  alphabet. 
To  be  sure,  Regiomontanus  and  Stifel  in  Germany,  and  Cardan  in 
Italy,  used  letters  before  him,  but  Vieta  extended  the  idea  and  first 
made  it  an  essential  part  of  algebra.  The  new  algebra  was  called  by 
him  logistica  speciosa  in  distinction  to  the  old  logislica  numerosa. 
Vieta's  formalism  differed  considerably  from  that  of  to-day.  The 
equation  a^+T,arb-{-T,ab'^-\-b^={a-\-hY  was  written  by  him  "a  cubus 
+b  in  a  quadr.  3-fa  in  b  quadr.  3+6  cubo  aequalia  a+b  cubo."  In 
numerical  equations  the  unknown  quantity  was  denoted  by  N,  its 
square  by  Q,  and  its  cube  by  C.  Thus  the  equation  x^  —  ^x^+idx-z^o 
was  written  i  C-SQ-(-i6  N  cequal.  40.  Vieta  used  the  term  "co- 
efficient," but  it  was  little  used  before  the  close  of  the  seventeenth 
century.^  Sometimes  he  uses  also  the  term  "polynomial."  Observe 
that  exponents  and  our  symbol  (  =  )  for  equality  were  not  yet  in  use; 
but  that  Vieta  employed  the  Maltese  cross  (-I-)  as  the  short-hand 
symbol  for  addition,  and  the  (  — )  for  subtraction.  These  two  char- 
acters had  not  been  in  very  general  use  before  the  time  of  Vieta.  "  Ic 
is  very  singular,"  says  Hallam,  "that  discoveries  of  the  greatest  con- 
venience, and,  apparently,  not  above  the  ingenuity  of  a  village  school- 
master, should  have  been  overlooked  by  men  of  extraordinary  acute- 
ness  like  Tartaglia,  Cardan,  and  L.  Ferrari;  and,  hardly  less  so  that,  by 
dint  of  that  acuteness,  they  dispensed  with  the  aid  of  these  contriv- 
ances in  which  we  suppose  that  so  much  of  the  utility  of  algebraic  ex- 
pression consists."  Even  after  improvements  in  notation  were  once 
proposed,  it  was  with  extreme  slowness  that  they  Avere  admitted  into 
general  use.  They  were  made  oftener  by  accident  than  design,  and 
their  authors  had  little  notion  of  the  effect  of  the  change  which  they 
were  making.  The  introduction  of  the  -|-  and  —  symbols  seems  to  be 
due  to  the  Germans,  who,  although  they  did  not  enrich  algebra  dur- 
ing the  Renaissance  with  great  inventions,  as  did  the  Italians,  still  cul- 
tivated it  with  great  zeal.  The  arithmetic  of  John  Widmann,  brought 
out  in  1489  in  Leipzig,  is  the  earliest  printed  book  in  which  the  +  and 
—  symbols  have  been  found.  The  +  sign  is  not  restricted  by  him  to 
ordinary  addition;  it  has  the  more  general  meaning  "et"  or  "and" 
as  in  the  heading,  "regula  augmenti  +  decrementi."  The  —  sign  is 
used  to  indicate  subtraction,  but  not  regularly  so.  The  word  'plus" 
does  not  occur  in  Widmann's  text;  the  word  "minus"  is  used  only  two 
or  three  times.     The  symbols  +  and  —  are  used  regularly  for  addi- 

1  EncydopSdic  dcs  sciences  malhimatiques ,  Tome  I,  Vol.  2,  1907,  p.  2. 


I40  A  HISTORY  OF  MATHEMATICS 

tion  and  subtraction,  in  1521/  in  the  arithmetic  of  Grammateus, 
(Heinrich  Schreiber,  died  1525)  a  teacher  at  the  University  of  Vienna. 
His  pupil,  Christoflf  Rudolff ,  the  writer  of  the  first  text-book  on  algebra 
in  the  German  language  (printed  in  1525),  employs  these  symbols  also. 
So  did  Stifel,  who  brought  out  a  second  edition  of  Rudolff's  Coss  in 
1553.  Thus,  by  slow  degrees,  their  adoption  became  universal.  Sev- 
eral independent  paleographic  studies  of  Latin  manuscripts  of  the 
fourteenth  and  fifteenth  centuries  make  it  almost  certain  that  the 
sign  +  comes  from  the  Latin  et,  as  it  was  cursively  written  in  manu- 
scripts just  before  the  time  of  the  invention  of  printing.^  The  ori- 
gin of  the  sign  —  is  still  uncertain.  There  is  another  short-hand 
symbol  of  which  we  owe  the  origin  to  the  Germans.  In  a  manu- 
script published  sometime  in  the  fifteenth  century,  a  dot  placed 
before  a  number  is  made  to  signify  the  extraction  of  a  root  of 
that  number.  This  dot  is  the  embryo  of  our  present  symbol  for  the 
square  root.  Christoff  Rudolff,  in  his  algebra,  remarks  that  "the 
radix  quadrata  is,  for  brevity,  designated  in  his  algorithm  with  the 
character  V,  as  V4."  Here  the  dot  has  grown  into  a  symbol  much 
like  our  own.  This  san-.e  symbol  was  used  by  Michael  Stifel.  Our 
sign  of  equality  is  due  to  Robert  Recorde  (1510-1558),  the  author  of 
The  Whetstone  of  Witte  (1557),  which  is  the  first  English  treatise  on 
algebra.  He  selected  this  symbol  because  no  two  things  could  be 
more  equal  than  two  parallel  lines  =.  The  si-^n  -^  for  division  was 
first  used  by  Johaiin  Heinrich  Rahn,  a  Swiss,  in  his  Teuische  Algebra, 
Zurich,  1659,  and  was  introduced  in  England  through  Thomas 
Brancker's  translation  of  Rahn's  book,  London,  1668. 

Michael  Stifel  (i4S6?-i567),  the  greatest  German  algebraist  of  the 
sixteenth  century,  was  born  in  Esslingen,  and  died  in  Jena.  He  was 
educated  in  the  monastery  of  his  native  place,  and  afterwards  be- 
came Protestant  minister.  The  study  of  the  significance  of  mystic 
numbers  in  Revelation  and  in  Daniel  drew  him  to  mathematics.  He 
studied  German  and  Italian  works,  and  published  in  1544,  in  Latin, 
a  book  entitled  Arithnietica  integra.  Melanchthon  wrote  a  preface  to 
it.  Its  three  parts  treat  respectively  of  rational  numbers,  irrational 
numbers,  and  algebra.  Stifel  gives  a  table  containing  the  numerical 
values  of  the  binomial  coefficients  for  powers  below  the  i8th.  He  ob- 
serves an  advantage  in  letting  a  geometric  progression  correspond  to 
an  arithmetical  progression,  and  arrives  at  the  designation  of  integral 
powers  by  numbers.  Here  are  the  germs  of  the  theory  of  exponents 
and  of  logarithms.  In  1545  Stifel  published  an  arithmetic  in  German. 
His  edition  of  Rudolff's  Coss  contains  rules  for  solving  cubic  equations, 
derived  from  the  v/ritings  of  Cardan. 

^  G.  Enestrom  in  Bihliotheca  mathematica,  3.  S.,  Vol.  9,  1908-09,  pp.  155-157; 
Vol.  14,  1914,  p.  27S. 

2  For  references  see  M.  Cantor,  op.  cit.,  Vol.  IF,  2.  Ed.,  1900,  p.  231;  J.  Tropfke, 
op.  cit.,  Vol.  I,  1902,  pp.  133,  134. 


THE  RENAISSANCE  141 

We  remarked  above  that  Vieta  discarded  negative  roots  of  equa- 
tions. Indeed,  we  find  few  algebraists  before  and  during  the  Renais- 
sance who  understood  the  significance  even  of  negative  quantities. 
Fibonacci  seldom  uses  them.  Pacioli  states  the  rule  that  "  minus  times 
minus  gives  plus,"  but  applies  it  really  only  to  the  development  of  the 
product  of  (a  —  b)  (c  —  d);  purely  negative  quantities  do  not  appear 
in  his  work.  The  German  "Cossist"  (algebraist),  Michael  Stifel, 
speaks  as  early  as  1544  of  numbers  which  are  "absurd"  or  "fictitious 
below  zero,"  and  which  arise  when  "real  numbers  abo\'e  zero"  are 
subtracted  from  zero.  Cardan,  at  last,  speaks  of  a  "pure  minus"; 
"but  these  ideas,"  says  H.  Hankel,  "remained  sparsely,  and  until 
the  beginning  of  the  seventeenth  century,  mathematicians  dealt  ex- 
clusively with  absolute  positive  quantities."  One  of  the  first  alge- 
braists who  occasionally  place  a  purely  negative  quantity  by  itself  on 
one  side  of  an  equation,  is  T.  Harriot  in  England.  As  regards  the  rec- 
ognition of  negative  roots,  Cardan  and  Bombelli  were  far  in  advance 
of  all  writers  of  the  Renaissance,  including  Vieta.  Yet  even  they 
mentioned  these  so-called  false  or  fictitious  roots  only  in  passing,  and 
without  grasping  their  real  significance  and  importance.  On  this 
subject  Cardan  and  Bombelli  had  advanced  to  about  the  same  point 
as  had  the  Hindu  Bhaskara,  who  saw  negative  roots,  but  did  not  ap- 
prove of  them.  The  generalization  of  the  conception  of  quantity  so 
as  to  include  the  negative,  was  an  exceedingly  slow  and  difficult  process 
in  the  development  of  algebra. 

We  shall  now  consider  the  history  of  geometry  during  the  Renais- 
sance. Unlike  algebra,  it  made  hardly  any  progress.  The  greatest 
gain  was  a  more  intimate  knowledge  of  Greek  geometry.  No  essen- 
tial progress  was  made  before  the  time  of  Descartes.  Regiomontanus, 
Xylander  (Wilhelm  Holzmann,  1532-1576)  of  Augsburg,  Tartaglia, 
Federigo  Commandino  (1509-1575)  of  Urbino  in  Italy,  Maurolycus 
and  others,  made  translations  of  geometrical  works  from  the  Greek. 
The  description  and  instrumental  construction  of  a  new  curve,  the 
epicycloid,  is  explained  by  Albrecht  Diirer  (1471-152S),  the  celebrated 
painter  and  sculptor  of  Niirnberg,  in  a  book,  Underweysung  der  Mes- 
sung  mil  dent  Zyrkel  und  rychtscheyd,  1525.  The  idea  of  such  a  curve 
goes  back  at  least  as  far  as  Hipparchus  who  used  it  in  his  astronomical 
theory  of  epicycles.  The  epicycloid  does  not  again  appear  in  history 
until  the  time  of  G.  Desargues  and  P.  La  Hire.  Diirer  is  the  earliest 
writer  in  the  Occident  to  call  attention  to  magic  squares.  A  simple 
magic  square  appears  in  his  celebrated  painting  called  "Melancholia." 

Johannes  Werner  (1468-1528)  of  Niirnberg  published  in  1522  the 
first  work  on  conies  which  appeared  in  Christian  Europe.  Unlike  the 
geometers  of  old,  he  studied  the  sections  in  relation  with  the  cone,  and 
derived  their  properties  directly  from  it.  This  mode  of  studying  the 
conies  was  followed  by  Franciscus  Maurolycus  (1494-1575)  of  Mes- 
sina.   The  latter  is,  doubtless,  the  greatest  geometer  of  the  sixteenth 


142  A  HISTORY  OF  MATHEMATICS 

century.  From  the  notes  of  Pappus,  he  attempted  to  restore  the  mi'ss^ 
ing  fifth  book  of  Apollonius  on  maxima  and  minima.  His  chief  work  is 
his  masterly  and  original  treatment  of  the  conic  sections,  wherein  he 
discusses  tangents  and  asymptotes  more  fully  than  Apollonius  had 
done,  and  applies  them  to  various  physical  and  astronomical  problems. 
To  Maurolycus  has  been  ascribed  also  the  discovery  of  the  inference 
by  mathematical  induction.^  It  occurs  in  his  introduction  to  his  Opus- 
cula  maihematica,  Venice,  1575.  Later,  mathematical  induction  was 
used  by  Pascal  in  his  Traite  du  triangle  arithmUique  (1662).  Processes 
akin  to  mathematical  induction,  some  of  which  would  yield  the  mod- 
ern mathematical  induction  by  introducing  some  slight  change  in  the 
mode  of  presentation  or  in  the  point  of  view,  were  given  before  Mau- 
rolycus. Giovanni  Campano  (latinized  form,  Campanus)  of  Novara 
in  Italy,  in  his  edition  of  Euclid  (1260),  proves  the  irrationality  of  the 
golden  section  by  a  recurrent  mode  of  inference  resulting  in  a  reductio 
ad  absurdum.  But  he  does  not  descend  by  a  regular  progression  from  n 
to  n—  I,  n—  2,  etc.,  but  leaps  irregularly  over,  perhaps,  several  integers. 
Campano's  process  was  used  later  by  Fermat.  A  recurrent  mode  of 
inference  is  found  in  Bhaskara's  "cyclic  method"  of  solving  inde- 
terminate equations,  in  Theon  of  Smyrna  (about  130  a.  d.)  and  in 
Proclus's  process  for  finding  numbers  representing  the  sides  and  di- 
agonals of  squares;  it  is  found  in  Euclid's  proof  {Elements  IX,  20)  that 
the  number  of  primes  is  infinite. 

The  foremost  geometrician  of  Portugal  was  Pedro  Nunes^  (1502- 
1578)  or  Nonius.  He  showed  that  a  ship  sailing  so  as  to  make  equal 
angles  with  the  meridians  does  not  travel  in  a  straight  line,  nor  usually 
along  the  arc  of  a  great  circle,  but  describes  a  path  called  the  loxo- 
dromic  curve.  Nunes  invented  the  "nonius"  and  described  it  in 
his  De  crepuscidis,  Lisbon,  1542.  It  consists  in  the  juxtaposition  of 
equal  arcs,  one  arc  divided  into  m  equal  parts  and  the  other  into  m+i 
equal  parts.  Nonius  took  m=^g.  The  instrument  is  also  called 
a  "vernier,"  after  the  Frenchman  Pierre  Vernier,  who  re-invented  it 
in  163 1.  The  foremost  French  mathematician  before  Vieta  was  Peter 
Ramus  (1515-1572),  who  perished  in  the  massacre  of  St.  Bartholomew. 
Vieta  possessed  great  familiarity  with  ancient  geometry.  The  new 
form  which  he  gave  to  algebra,  by  representing  general  quantities  by 
letters,  enabled  him  to  point  out  more  easily  how  the  construction  of 
the  roots  of  cubics  depended  upon  the  celebrated  ancient  problems  of 
the  duplication  of  the  cube  and  the  trisection  of  an  angle.  He  reached 
the  interesting  conclusion  that  the  former  problem  includes  the  solu- 
tions of  all  cubics  in  which  the  radical  in  Tartaglia's  formula  is  real, 
but  that  the  latter  problem  includes  only  those  leading  to  the  irredu- 
cible case 

iQ.  Vacca  in  Bulletin  Am.  Math.  Society,  2.  S.,  Vol.  16.  1909,  f  70.  See  also 
F.  Cajori  in  Vol.  15,  pp.  407-409. 

2  See  R.  Guimaraes,  Pedro  Nunes,  Coimpre,  1915. 


THE  RENAISSANCE 


143 


The  problem  of  the  quadrature  of  the  circle  was  revived  in  this  age, 
nnd  was  zealously  studied  even  by  men  of  eminence  and  mathematical 
ability.  The  army  of  circle-squarers  became  most  formidable  during 
the  seventeenth  centur\'.  Among  the  first  to  revive  this  problem  was 
the  German  Cardinal  Nicolaus  Cusanus  (1401-1464),  who  had  the 
reputation  of  being  a  great  logician.  His  fallacies  were  exposed  to 
full  view  by  Regiomontanus.  As  in  this  case,  so  in  others,  every  quad- 
rator  of  note  raised  up  an  opposing  mathematician:  Oronce  Fine  was 
met  by  Jean  Buteo  (c.  1492-1572)  and  P.  Nunes;  Joseph  Scaliger  by 
Vieta,  Adrianus  Romanus,  and  Clavius;  a  Quercu  by  Adriaen  An- 
thonisz  (1527-1607).  Two  mathematicians  of  Netherlands,  Adrianus 
Romanus  (1561-1615)  and  Ludolph  van  Ceulen  (1540-1610),  occu- 
pied themselves  with  approximating  to  the  ratio  between  the  circumfer- 
ence and  the  diameter.  The  former  carried  the  value  tt  to  15,  the  lat- 
ter to  35,  places.  The  value  of  tt  is  therefore  often  named  "Ludolph's 
number."  His  performance  was  considered  so  extraordinary,  that  the 
numbers  were  cut  on  his  tomb-stone  (now  lost)  in  St.  Peter's  church- 
yard, at  Leyden.  These  men  had  used  the  Archimedian  method  of 
in-  and  circum-scribed  polygons,  a  method  refined  in  162 1  by  Wille- 
broni  Sndlins  (i 580-1626)  who  showed  how  narrower  limits  may  be 
obtained  for  7r  without  increasing  the  number  of  sides  of  the  poly- 
gons. Snellius  used  two  theorems  equivalent  to  \  (2  sin  6  tan  B)  Z  d/. 
3/(2  csc^+cot^).  The  greatest  refinements  in  the  use  of  the  geo- 
metrical method  of  Archimedes  were  reached  by  C.  Huyghens  in  his 
De  circidi  niagnitiidine  inventa,  1654,  and  by  James  Gregory  (1638- 
1675),  professor  at  St.  Andrews  and  Edinburgh,  in  his  Exercilationes 
geomctriccE,  1668,  and  Vera  circiiU  et  hyperbolae  quadratura,  1667. 
Gregory  gave  several  formulas  for  approximating  to  tt  and  in  the 
second  of  these  publications  boldly  attempted  to  prove  by  the  Ar- 
chimedean algorithm  that  the  quadrature  of  the  circle  is  impossible. 
Huyghens  showed  that  Gregory's  proof  is  not  conclusive,  although 
he  himself  believed  that  the  quadrature  is  impossible.  Other  attempts 
to  prove  this  impossibility  were  made  by  Thomas  Fautat  De  Lagny 
(1660-1734)  of  Paris,  in  1727,  Joseph  Saurin  (1659-1737)  in  1720, 
Isaac  Newton  in  his  Principia  I,  6,  lemma  28,  E.  Waring,  L.  Euler, 
1771. 

That  these  proofs  would  lack  rigor  was  almost  to  be  expected,  as 
long  as  no  distinction  was  made  between  algebraical  and  transcen- 
dental numbers. 

The  earliest  explicit  expression  for  tt  by  an  infinite  number  of  op- 
erations was  found  by  Vieta.  Considering  regular  polygons  of  4,  8, 
16,  .  .  .  sides,  inscribed  in  a  circle  of  unit  radius,  he  found  that  the 
area  of  the  circle  is 


^Wh-^\H\\+hVh+h^2 


144  A  HISTORY  OF  MATHEMATICS 

from  which  we  obtain 

"""_      _        ^  ,  which  may  be  derived  from  Euler's  formula  ^ 

B=—ir-^ J-! ■  '  (^/tt),  by  taking  0=7r/2. 

COS^/2  COS^/4  COS^/8  ... 

As  mentioned  earlier,  it  was  Adrianus  Romanus  (1561-1615)  of  Lou- 
vain  who  propounded  for  solution  that  equation  of  the  forty-fifth  degree 
solved  by  Vieta.  On  receiving  Vieta's  solution,  he  at  once  departed  for 
Paris,  to  make  his  acquaintance  with  so  great  a  master.  Vieta  proposed 
to  him  the  Apollonian  problem,  to  draw  a  circle  touching  three  given 
circles.  "Adrianus  Romanus  solved  the  problem  by  the  intersection  of 
two  hyperbolas;  but  this  solution  did  not  possess  the  rigor  of  the  ancient 
geometry.  Vieta  caused  him  to  see  this,  and  then,  in  his  turn,  pre- 
sented a  solution  which  had  all  the  rigor  desirable."  -  Romanus 
did  much  toward  simplifying  spherical  trigonometry  by  reducing,  by 
means  of  certain  projections,  the  28  cases  in  triangles  then  considered 
to  only  six. 

Mention  must  here  be  made  of  the  improvements  of  the  Julian 
calendar.  The  yearly  determination  of  the  movable  feasts  had  for 
a  long  time  been  connected  with  an  untold  amount  of  confusion.  The 
rapid  progress  of  astronomy  led  to  the  consideration  of  this  subject, 
and  many  new  calendars  were  proposed.  Pope  Gregory  XIII  con- 
voked a  large  number  of  mathematicians,  astronomers,  and  prelates, 
who  decided  upon  the  adoption  of  the  calendar  proposed  by  the  Jesuit 
Christophonis  Clavius  (1537-1612)  of  Rome.  To  rectify  the  errors  of 
the  Julian  calendar  it  was  agreed  to  write  in  the  new  calendar  the  15th 
of  October  immediately  after  the  4th  of  October  of  the  year  1582. 
The  Gregorian  calendar  met  with  a  great  deal  of  opposition  both 
among  scientists  and  among  Protestants.  Clavius,  who  ranked  high 
as  a  geometer,  met  the  objections  of  the  former  most  ably  and  effec- 
tively; the  prejudices  of  the  latter  passed  away  with  time. 

The  passion  for  the  study  of  mystical  properties  of  numbers  de- 
scended from  the  ancients  to  the  moderns.  Much  was  written  on 
numerical  mysticism  even  by  such  eminent  men  as  Pacioli  and  Stifel. 
The  Numerorum  Mysferia  of  Peter  Bungus  covered  700  quarto  pages. 
He  worked  with  great  industry  and  satisfaction  on  666,  which  is  the 
number  of  the  beast  in  Revelation  (xiii,  18),  the  symbol  of  Antichrist. 
He  reduced  the  name  of  the  "impious"  Martin  Luther  to  a  form  which 
may  express  this  formidable  number.  Placing  a=i,  h=2,  etc.,  ^=10, 
/=  20,  etc.,  he  finds,  after  misspelling  the  name,  that  M(30)A(])R(8o)T(ioo) 
I(9)N(40)L(2o)V(2oo)T(ioo)E(5)R(80)A(i)  constitutes  the  number  required. 
These  attacks  on  the  great  reformer  were  not  unprovoked,  for  his 

1  E.  W.  Hobson,  Squaring  the  Circle,  Cambridge,  1913,  pp.  26,  27,  31. 

2  A.  Quetelet,  Hisloire  des  Sciences  mathematiques  el  physiques  ekes  les  Beiges. 
Bruxelles,  1S64,  p.  137. 


VIETA  TO  DESCARTES  145 

friend.  Michael  Stifel,  the  most  acute  and  original  of  the  early  mathe- 
maticians of  Germany,  exercised  an  equal  ingenuity  in  showing  that 
the  above  number  referred  to  Pope  Leo  X, — a  demonstration  which 
gave  Stifel  unspeakable  comfort.^ 

Astrology  also  was  still  a  favorite  study.  It  is  well  known  that  Car- 
dan, Maurolycus,  Regiomontanus,  and  many  other  eminent  scientists 
who  lived  at  a  period  even  later  than  this,  engaged  in  deep  astrological 
study;  but  it  is  not  so  generally  known  that  besides  the  occult  sciences 
already  named,  men  engaged  in  the  mystic  study  of  star-polygons 
and  magic  squares.  ''The  pentagramma  gives  you  pain,"  says  Faust 
to  Mephistophcles.  It  is  of  deep  psychological  interest  to  see  scientists, 
like  the  great  Kepler,  demonstrate  on  one  page  a  theorem  on  star- 
polygons,  with  strict  geometric  rigor,  while  on  the  next  page,  perhaps, 
he  explains  their  use  as  amulets  or  in  conjurations.  Playfair,  speaking 
of  Cardan  as  an  astrologer,  calls  him  "a  melancholy  proof  that  tliere 
is  no  folly  or  weakness  too  great  to  be  united  to  high  intellectual  at- 
tainments." -  Let  our  judgment  not  be  too  harsh.  The  period  under 
consideration  is  too  near  the  Middle  Ages  to  admit  of  complete  eman- 
cipation from  mysticism  even  among  scientists.  Scholars  like  Kepler, 
Napier,  Albrecht  Diirer,  while  in  the  van  of  progress  and  planting 
one  foot  upon  the  firm  ground  of  truly  scientific  inquiry,  were  still 
resting  with  the  other  foot  upon  the  scholastic  ideas  of  preceding  ages. 

Viefa  to  Descartes 

The  ecclesiastical  power,  which  in  the  ignorant  ages  was  an  unmixed 
benefit,  in  more  enlightened  ages  became  a  serious  evil.  Thus,  in 
France,  during  the  reigns  preceding  that  of  Henry  IV,  the  theological 
spirit  predominated.  This  is  painfully  shown  by  the  massacres  of 
Vassy  and  of  St.  Bartholomew.  Being  engaged  in  religious  disputes, 
people  had  no  leisure  for  science  and  for  secular  literature.  Hence, 
down  to  the  time  of  Henry  IV,  the  French  "had  not  put  forth  a  single 
work,  the  destruction  of  which  would  now  be  a  loss  to  Europe."  In 
England,  on  the  other  hand,  no  religious  wars  were  waged.  The  people 
were  comparatively  indifferent  about  religious  strifes;  they  concen- 
trated their  ability  upon  secular  matters,  and  acquired,  in  the  six- 
teenth century,  a  literature  which  is  immortalized  by  the  genius  of 
Shakespeare  and  Spenser.  This  great  literary  age  in  England  was 
followed  by  a  great  scientific  age.  At  the  close  of  the  skteenth  cen- 
tury, the  shackles  of  ecclesiastical  authority  were  thrown  off  by  France. 
The  ascension  of  Henry  IV  to  the  throne  was  followed  in  1598  by  the 
Edict  of  Nantes,  granting  freedom  of  worship  to  the  Huguenots,  and 
thereby  terminating  religious  wars.    The  genius  of  the  French  nation 

'  G.  Peacock,  op.  cil.,  p.  424. 

^John  Playfair,  "Progress  of  the  Mathematical  and  Ph)'sical  Sciences"  in  En- 
cyclopaedia Brilannka.  7th  ed.,  continued  in  8th  Ed.,  by  Sir  John  Leslie. 


146  A  HIS  TORY  OF  MATHEMAT.CS 

now  began  to  blossom.  Cardinal  Richelieu,  during  the  rf\gn  of  Louis 
XIII,  pursued  the  broad  policy  of  not  favoring  the  opinions  of  any 
sect,  but  of  promoting  the  interests  of  the  nation.  His  age  was  re- 
markable for  the  progress  of  knowledge.  It  produced  that  great  secu- 
lar literature,  the  counterpart  of  which  was  found  in  England  in  the 
sixteenth  century.  The  seventeenth  century  was  made  illustrious 
also  by  the  great  French  mathematicians,  Roberval,  Descartes,  Des- 
argues,  Fermat,  and  Pascal. 

More  gloomy  is  the  picture  in  Germany.  The  great  changes  which 
revolutionized  the  world  in  the  sixteenth  century,  and  which  led  Eng- 
land to  national  greatness,  led  Germany  to  degradation.  The  first 
effects  of  the  Reformation  there  were  salutary.  At  the  close  of  the 
fifteenth  and  during  the  sixteenth  century,  Germany  had  been  con- 
spicuous for  her  scientific  pursuits.  She  had  been  a  leader  in  as- 
tronomy and  trigonometry.  Algebra  also,  excepting  for  the  discoveries 
in  cubic  equations,  was,  before  the  time  of  Vieta,  in  a  more  advanced 
state  there  than  elsewhere.  But  at  the  beginning  of  the  seventeenth 
century,  when  the  sun  of  science  began  to  rise  in  France,  it  set  in  Ger- 
many. Theologic  disputes  and  religious  strife  ensued.  The  Thirty 
Years'  War  (1618-1648)  proved  ruinous.  The  German  empire  was 
shattered,  and  became  a  mere  lax  confederation  of  petty  despotisms. 
Commerce  was  destroyed;  national  feeling  died  out.  Art  disappeared, 
and  in  hterature  there  was  only  a  slavish  imitation  of  French  arti- 
ficiality. Nor  did  Germany  recover  from  this  low  state  for  200  years; 
for  in  1756  began  another  struggle,  the  Seven  Years'  War,  which 
turned  Prussia  into  a  wasted  land.  Thus  it  followed  that  at  the  be- 
ginning of  the  seventeenth  century,  the  great  Kepler  was  the  only 
German  mathematician  of  eminence,  and  that  in  the  interval  of  200 
years  between  Kepler  and  Gauss,  there  arose  no  great  mathematician 
in  Germany  excepting  Leibniz. 

Up  to  the  seventeenth  century,  mathematics  was  cultivated  but  little 
in  Great  Britain.  During  the  sixteenth  century,  she  brought  forth 
no  mathematician  comparable  with  Vieta,  Stife'l,  or  Tartaglia.  But 
with  the  time  of  Recorde,  the  English  became  conspicuous  for  numeri- 
cal skill.  The  first  important  arithmetical  work  of  English  authorship 
was  published  in  Latin  in  1522  by  CuthbertTonstall  (1474-1559).  He 
had  studied  at  Oxford,  Cambridge,  and  Padua,  and  drew  freely  from 
the  works  of  Pacioli  and  Regiomontanus.  Reprints  of  his  arithmetic 
appeared  in  England  and  France.  After  Recorde  the  higher  branches 
of  mathematics  began  to  be  studied.  Later,  Scotland  brought  forth 
John  Napier,  the  inventor  of  logarithms.  The  instantaneous  appre- 
ciation of  their  value  is  doubtless  the  result  of  superiority  in  calcula- 
tion. In  Italy,  and  especially  in  France,  geometry,  which  for  a  long 
time  had  been  an  almost  stationary  science,  began  to  be  studied  with 
success.  Galileo,  Torricelli,  Roberval,  Fermat,  Desargues,  Pascal, 
Descartes,  and  the  English  Wallis  are  the  great  revolutioners  of  this 


VIETA  TO  DESCARTES  147 

science.  Theoretical  mechanics  began  to  be  stiKiied.  The  foundations 
were  laid  by  Fermat  and  Pascal  for  the  theory  of  numbers  and  the 
theory  of  probability. 

We  shall  first  consider  the  improvements  made  in  the  art  of  calcu- 
lating. The  nations  of  antiquity  experimented  thousands  of  years 
upon  numeral  notations  before  they  happened  to  strike  upon  the  so- 
called  "Arabic  notation."  In  the  simple  expedient  of  the  cipher, 
which  was  permanently  introduced  by  the  Hindus,  mathematics  re- 
ceived one  of  the  most  powerful  impulses.  It  would  seem  that  after 
the  "Arabic  notation"  was  once  thoroughly  understood,  decimal 
fractions  would  occur  at  once  as  an  obvious  extension  of  it.  But  "it 
is  curious  to  think  how  much  science  had  attempted  in  physical  re- 
search and  how  deeply  numbers  had  been  pondered,  before  it  was  per- 
ceived that  the  all-powerful  simplicity  of  the  '  Arabic  notation '  was  as 
valuable  and  as  manageable  in  an  infinitely  descending  as  in  an  in- 
finitely ascending  progression."  ^  Simple  as  decimal  fractions  ap- 
pear to  us,  the  invention  of  them  is  not  the  result  of  one  mind  or  even 
of  one  age.  They  came  into  use  by  almost  imperceptible  degrees.  The 
first  mathematicians  identified  with  their  history  did  not  perceive 
their  true  nature  and  importance,  and  failed  to  invent  a  suitable  no- 
tation. The  idea  of  decimal  fractions  makes  its  first  appearance  in 
methods  for  approximating  to  the  square  roots  of  numbers.  Thus 
John  of  Seville,  presumably  in  imitation  of  Hindu  rules,  adds  2n  ci- 
phers to  the  number,  then  finds  the  square  root,  and  takes  this  as  the 
numerator  of  a  fraction  whose  denominator  is  i  followed  by  n  ciphers. 
The  same  method  was  followed  by  Cardan,  but  it  failed  to  be  generally 
adopted  even  by  his  Italian  contemporaries;  for  otherwise  it  would 
certainly  have  been  at  least  mentioned  by  Pietro  Cataldi  (died  1626) 
in  a  work  devoted  exclusively  to  the  extraction  of  roots.  Cataldi, 
and  before  him  Bombelli  in  1572,  find  the  square  root  by  means  of 
continued  fractions — a  method  ingenious  and  novel,  but  for  practical 
purposes  inferior  to  Cardan's.  Oronce  Fine  (14Q4-1555)  in  France 
(called  also  Orontius  Finaeus),  and  William  Buckley  (died  about 
1550)  in  England  extracted  the  square  root  in  the  same  way  as 
Cardan  and  John  of  Seville.  The  invention  of  decimals  has  been 
frequently  attributed  to  Regiomontanus,  on  the  ground  that  in- 
stead of  placing  the  sinus  totus,  in  trigonometry,  equal  to  a  multiple 
of  60,  like  the  Greeks,  he  put  it=  100,000.  But  here  the  trigonomet- 
rical lines  were  expressed  in  integers,  and  not  in  fractions.  Though 
he  adopted  a  decimal  division  of  the  radius,  he  and  his  successors 
did  not  apply  the  idea  outside  of  trigonometry  and,  indeed,  had  no 
notion  whatever  of  decimal  fractions.  To  Simon  Stevin  (1548- 
1620)  of  Bruges  in  Belgium,  a  man  who  did  a  great  deal  of  work  in 
most  diverse  fields  of  science,  we  owe  the  first  systematic  treatment  of 
decimal  fractions.  In  his  La  Disme  (1585)  he  describes  in  very  express 
'  Mark  Napier,  Memoirs  of  John  Napier  of  Merchiston.    Edinburgh,  1834. 


148  A  HISTORY  OF  MATHEMATICS 

terms  the  advantages,  not  only  of  decimal  fractions,  but  also  of  the 
decimal  division  in  systems  of  weights  and  measures.  Stevin  applied 
the  new  fractions  "to  all  the  operations  of  ordinary  arithmetic."  ^ 
What  he  lacked  was  a  suitable  notation.  In  place  of  our  decimal  point, 
he  used  a  cipher;  to  each  place  in  the  fraction  was  attached  the  cor- 
responding index.  Thus,  in  his  notation,  the  number  5.912  would  be 
0123 

5912  or  509©i®20.  These  indices,  though  cumbrous  in  practice,  are 
of  interest,  because  they  embody  the  notion  of  powers  of  numbers. 
Stevin  considered  also  fractional  powers.  He  says  that  "|"  placed 
within  a  circle  would  mean  x'1%  but  he  does  not  actually  use  his  nota- 
tion. This  notion  had  been  advanced  much  earlier  by  Oresme,  but 
it  had  remained  unnoticed.  Stevin  found  the  greatest  common  di- 
visor of  x^+x"^  and  x^+jx+G  by  the  process  of  continual  division, 
thereby  applying  to  polynomials  Euclid's  mode  of  finding  the  greatest 
common  divisor  of  numbers,  as  explained  in  Book  VII  of  his  Elements. 
Stevin  was  enthusiastic  not  only  over  decimal  fractions,  but  also  over 
the  decimal  division  of  weights  and  measures.  He  considered  it  the 
duty  of  governments  to  establish  the  latter.  He  advocated  the  deci- 
mal subdivision  of  the  degree.  No  improvement  was  made  in  the 
notation  of  decimals  till  the  beginning  of  the  seventeenth  century. 
After  Stevin,  decimals  were  used  by  Joost  Biirgi  (1552-1632),  a  Swiss 
by  birth,  who  prepared  a  manuscript  on  arithmetic  soon  after  1592,  and 
by  Johann  Hartmann  Beyer,  who  assumes  the  invention  as  his  own. 
In  1603,  he  published  at  Frankfurt  on  the  Main  a  Logistica  Decimalis. 
Historians  of  mathematics  do  not  yet  agree  to  whom  the  first  intro- 
duction of  the  decimal  point  or  comma  should  be  ascribed.  Among 
.the  candidates  for  the  honor  are  Pellos  (1492),  Biirgi  (1592),  Pitiscus 
(1608,  1612),  Kepler  (1616),  Napier  (1616,  1617).  This  divergence 
of  opinion  is  due  mainly  to  different  standards  of  judgment.  If  the 
requirement  made  of  candidates  is  not  only  that  the  decimal  point  or 
comma  was  actually  used  by  them,  but  that  they  must  give  evidence 
that  the  numbers  used  were  actually  decimal  fractions,  that  the  point 
or  comma  was  with  them  not  merely  a  general  symbol  to  indicate 
a  separation,  that  they  must  actually  use  the  decimal  point  in  opera- 
tions including  multiplication  or  division  of  decimal  fractions,  then 
it  would  seem  that  the  honor  falls  to  John  Napier,  who  exhibits  such 
use  in  his  Rabdologia,  161 7.  Perhaps  Napier  received  the  suggestion 
for  this  notation  from  Pitiscus  who,  according  to  G.  Enestrom,^  uses 
the  point  in  his  Trigonometria  of  1608  and  161 2,  not  as  a  regular  deci- 
mal point,  but  as  a  more  general  sign  of  separation.  Napier's  decimal 
point  did  not  meet  with  immediate  adoption.  W.  Oughtred  in  163 1 
designates  the  fraction  .56  thus,  0156.  Albert  Girard,  a  pupil  of  Stevin, 
in  1629  uses  the  point  on  one  occasion.    John  Wallis  in  1657  writes 

1  A.  Quetelet,  op.  cit.,  p.  158. 

-  Bibliotheca  malhcmalka,  3.  S.,  Vol.  6,  1905,  p.  109. 


VI ETA  TO  DESCARTES  149 

12I345,  but  afterwards  in  his  algebra  adopts  the  usual  point.  A.  De 
Morgan  says  that  ''to  the  first  quarter  of  the  eighteenth  century  we 
must  refer  not  only  the  complete  and  final  victory  of  the  decimal  point, 
but  also  that  of  the  now  universal  method  of  performing  the  operations 
of  division  and  extraction  of  the  square  root."  ^  We  have  dwelt  at 
some  length  on  the  progress  of  the  decimal  notation,  because  "the 
history  of  language  ...  is  of  the  highest  order  of  interest,  as  well  as 
utiUty :  its  suggestions  are  the  best  lesson  for  the  future  which  a  reflect- 
ing mind  can  have." 

The  miraculous  powers  of  modern  calculation  are  due  to  three  in- 
ventions: the  Arabic  Notation,  Decimal  Fractions,  and  Logarithms. 
The  invention  of  logarithms  in  the  first  quarter  of  the  seventeenth 
century  was  admirably  timed,  for  Kepler  was  then  examining  plane- 
tary orbits,  and  Galileo  had  just  turned  the  telescope  to  the  stars. 
During  the  Renaissance  German  mathematicians  had  constructed 
trigonometrical  tables  of  great  accuracy,  but  its  greater  precision 
enormously  increased  the  work  of  the  calculator.  It  is  no  exaggera- 
tion to  say  that  the  invention  of  logarithms  "  by  shortening  the  labors 
doubled  the  life  of  the  astronomer."  Logarithms  were  invented  by 
John  Napier  (1550-1617),  Baron  of  IMerchiston,  in  Scotland.  It  is 
one  of  the  greatest  curiosities  of  tiie  history  of  science  that  Napier 
constructed  logarithms  before  exponents  were  used.  To  be  sure, 
Stifel  and  Stevin  made  some  attempts  to  denote  powers  by  indices, 
but  this  notation  was  not  generally  known, — not  even  to  T.  Harriot, 
whose  algebra  appeared  long  after  Napier's  death.  That  logarithms 
flow  naturally  from  the  exponential  symbol  was  not  observed  until 
much  later.    What,  then,  was  Napier's  line  of  thought? 

Let  ^5  be  a  definite  line,  DE  a  line  extending  from  D  indefinitely. 
Imagine  two  points  starting  at  the  same  moment;  the  one  moving 

AC  B 

I I 


D  F  E 

from  A  toward  B,  the  other  from  D  toward  E.  Let  the  velocity  during 
the  first  moment  be  the  same  for  both:  let  that  of  the  point  on  line  DE 
be  uniform;  but  the  velocity  of  the  point  on  AB  decreasing  in  such 
a  way  that  when  it  arrives  at  any  point  C,  its  velocity  is  proportional 
to  the  remaining  distance  BC.  While  the  first  point  moves  over  a  dis- 
tance AC,  the  second  one  moves  over  a  distance  DF.  Napier  calls 
DF  the  logarithm  of  BC. 

He  first  sought  the  logarithms  only  of  sines;  the  line  AB  was  the 
sine  of  90°  and  was  taken  =10'';  BC  was  the  sine  of  the  arc,  and 

'  A.  De  Morgan,  Arithmetical  Books  from  the  Invention  of  Prinlins,  to  the  Present 
Time  Txjndon,  1847,  p.  xxvii. 


I50  A  HISTORY  OF  MATHEMATICS 

DF  its  logarithm.  We  notice  that  as  the  motion  proceeds^  BC 
decreases  in  geometrical  progression,  while  DF  increases  in  arith- 
metical   progression.      Let    AB  =  a=i6' ,    let    x=DF,    y=BC,    then 

AC=a—y.     The  velocity  of  the  point  C  is  — jr^=y;  this  gives 

-  nat.  log  y=t+c.    When  /=o,  then  y=a  and  c=-  nat.  log  a.    Again, 

let  T-  =  «  be  the  velocity  of  the  point  F,  then  x=at.    Substituting  for 

/  and  c  their  values  and  remembering  that  a=  lo'^  and  that  by  defini- 
tion a;=Nap.  log  y,  we  get 

Nap,  log  y=  lo^  nat.  log  — . 

It  is  evident  from  this  formula  that  Napier's  logarithms  are  not  the 
same  as  the  natural  logarithms.  Napier's  logarithms  increase  as  the 
number  itself  decreases.  He  took  the  logarithm  of  sin  90°= o;  i.  e. 
the  logarithm  of  10''= o.  The  logarithm  of  sin  a  increased  from  zero 
as  a  decreased  from  90°.  Napier's  genesis  of  logarithms  from  the  con- 
ception of  two  flowing  points  reminds  us  of  Newton's  doctrine  of 
fluxions.  The  relation  between  geometric  and  arithmetical  progres- 
sions, so  skilfully  utilized  by  Napier,  had  been  observed  by  Archi- 
medes, Stifel,  and  others.  What  was  the  base  of  Napier's  system  of 
logarithms?  To  this  we  reply  that  not  only  did  the  notion  of  a 
"base"  never  suggest  itself  to  him,  but  it  is  inapplicable  to  his 
system.  This  notion  demands  that  zero  be  the  logarithm  of  i;  in 
Napier's  system,  zero  is  the  logarithm  of  10^.  Napier's  great  in- 
vention was  given  to  the  world  in  1614  in  a  work  entitled  Mirificl 
logarithmorum  canonis  descriptio.  In  it  he  explained  the  nature  of 
his  logarithms,  and  gave  a  logarithmic  table  of  the  natural  sines  of 
a  quadrant  from  minute  to  minute.  In  16 19  appeared  Napier's 
Mirifici  logarithmorum  canonis  conslrudio,  as  a  posthumous  work,  in 
which  his  method  of  calculating  logarithms  is  explained.  An  Enghsh 
translation  of  the  Construdio,  by  W.  R.  Macdonald,  appeared  in 
Edinburgh,  in  1889. 

Henry  Briggs  (1556-1631),  in  Napier's  time  professor  of  geometry 
at  Gresham  College,  London,  and  afterwards  professor  at  Oxford, 
was  so  struck  with  admiration  of  Napier's  book,  that  he  left  his  studies 
in  London  to  do  homage  to  the  Scottish  philosopher.  Briggs  was  de- 
layed in  his  journey,  and  Napier  complained  to  a  common  friend,  "Ah, 
John,  Mr.  Briggs  will  not  come."  At  that  very  moment  knocks  were 
heard  at  the  gate,  and  Briggs  was  brought  into  the  lord's  chamber. 
Almiost  one-quarter  of  an  hour  was  spent,  each  beholcimg  the  other 
without  speaking  a  word.  At  last  Briggs  began:  "My  lord.  ^  have 
undertaken  this  long  journey  purposely  to  see  your  pers-Tin,  and  to 
know  by  what  engine  of  wit  or  ingenuity  you  came  first  to  think  of 


VIETA  TO  DESCARTES  151 

this  most  excellent  help  in  astronomy,  viz.  the  logarithms;  but,  m} 
lord,  being  by  you  found  out,  I  wonder  nobody  found  it  out  before, 
when  now  known  it  is  so  easy."  Briggs  suggested  to  Napier  the  ad- 
vantage that  would  result  from  retaining  zero  for  the  logarithm  of  the 
whole  sine,  but  choosing  10,000,000,000  for  the  logarithm  of  the  loth 
part  of  that  same  sine,  ;'.  c.  of  5°  44'  22".  Napier  said  that  he  had  al- 
ready thought  of  the  change,  and  he  pointed  out  a  slight  improvement 
on  Briggs'  idea;  viz.  that  zero  should  be  the  logarithm  of  i,  and 
10,000,000,000  that  of  the  whole  sine,  thereby  making  the  character- 
istic of  numbers  greater  than  unity  positive  and  not  negative,  as  sug- 
gested by  Briggs.  Briggs  admitted  this  to  be  more  convenient.  The 
invention  of  "Briggian  logarithms"  occurred,  therefore,  to  Briggs 
and  Napier  independently.  The  great  practical  advantage  of  the  new 
system  was  that  its  fundamental  progression  was  accommodated  to 
the  base,  10,  of  our  numerical  scale.  Briggs  devoted  all  his  energies 
to  the  construction  of  tables  upon  the  new  plan.  Napier  died  in  161 7, 
with  the  satisfaction  of  having  found  in  Briggs  an  able  friend  to  bring 
to  completion  his  unfinished  plans.  In  1624  Briggs  pubhshed  his 
Arithmetica  logarillmiica,  containing  the  logarithms  to  14  places  of 
numbers,  from  i  to  20,000  and  from  go,ooo  to  100,000.  The  gap  from 
20,000  to  90,000  was  filled  up  by  that  illustrious  successor  of  Napier 
and  Briggs,  Adrian  Vlacq  (i6oo?-i667).  He  was  born  at  Gouda  in 
Holland  and  livea  ten  years  in  London  as  a  bookseller  and  pubhsher. 
Being  driven  out  by  London  bookdealers,  he  settled  in  Paris  where  he 
met  opposition  again,  for  selling  foreign  books.  He  died  at  The  Hague. 
John  Milton,  in  his  Defensio  secunda,  published  an  abuse  of  him. 
Vlacq  published  in  16 28  a  table  of  logarithms  from  i  to  100,000,  of 
which  70,000  were  calculated  by  himself.  The  first  publication  of 
Briggian  logarithms  of  trigonometric  functions  was  made  in  1620  by 
Edmund  Gunter  (15S1-1626)  of  London,  a  colleague  of  Briggs,  who 
found  the  logarithmic  sines  and  tangents  for  every  minute  to  seven 
places.  Gunter  was  the  inventor  of  the  words  cosine  and  cotangent 
(1620). 

The  word  cosine  was  an  abbreviation  of  complemental  sine.  The 
invention  of  the  words  tangent  and  secant  is  due  to  the  physician  and 
mathematician,  Thomas  Finck,  a  native  of  Flensburg,  who  used  them 
in  his  Geometria  rotundi,  Basel,  15S3.  Gunter  is  known  to  engineers 
for  his  "Gunter's  chain."  It  is  told  of  him  that  "When  he  was  a  stu- 
dent at  Christ  College,  it  fell  to  his  lot  to  preach  the  Passion  sermon, 
which  some  old  divines  that  I  knew  did  hear,  but  they  said  that  it 
was  said  of  him  then  in  the  University  that  our  Savior  never  suffered 
so  much  since  his  passion  as  in  that  sermon,  it  was  such  a  lamented 
one."  ^  Briggs  devoted  the  last  years  of  his  life  to  calculating  more 
extensive  Briggian  logarithms  of  trigonometric  functions,  but  he  died 
in  163 1,  leaving  his  work  unfinished.  It  was  carried  on  by  Henry  Gel- 
^  Aubrey's  Brief  Lives,  Edition  A.  Clark,  1898,  Vol.  I,  p.  276. 


152  A  HISTORY  OF  MATHEMATICS 

librand  (1597-1637)  of  Gresham  College  in  London,  and  then  pub- 
lished by  Vlacq  at  his  own  expense.  Briggs  divided  a  degree  into 
100  parts,  as  was  done  also  by  N.  Roe  in  1633,  W.  Oughtred  in  1657, 
John  Newton  in  1658,  but  owing  to  the  publication  by  Vlacq  of  trigo- 
nometrical tables  constructed  on  the  old  sexagesimal  division,  Briggs' 
innovation  did  not  prevail.  Briggs  and  Vlacq  published  four  funda- 
mental works,  the  results  of  which  have  not  been  superseded  by  any 
subsequent  calculations  until  very  recently. 

The  word  "characteristic,"  as  used  in  logarithms,  first  occurs  in 
Briggs'  Arithmetica  logarithmica,  1624;  the  word  "mantissa"  was  in- 
troduced by  John  WaUis  in  the  Latin  edition  of  his  Algebra,  1693, 
p.  41,  and  was  used  by  L.  Euler  in  his  IntroducHo  in  analysin  in  1748, 
p.  85. 

The  only  rival  of  John  Napier  in  the  invention  of  logarithms  was  the 
Swiss  Joost  Biirgi  (1552-1632).  He  published  a  table  of  logarithms, 
Arithmetische  und  Geomelrische  Progresstabulen,  Prague,  1620,  but  he 
conceived  the  idea  and  constructed  his  table  independently  of  Napier. 
He  neglected  to  have  it  published  until  Napier's  logarithms  were 
known  and  admired  throughout  Europe. 

Among  the  various  inventions  of  Napier  to  assist  the  memory  of 
the  student  or  calculator,  is  "Napier's  rule  of  circular  parts"  for  the 
solution  of  spherical  right  triangles.  It  is,  perhaps,  "the  happiest 
example  of  artificial  memory  that  is  known."  Napier  gives  in  the 
Descriptio  a  proof  of  his  rule;  proofs  were  given  later  by  Johann 
Heinrich  Lambert  (1765)  and  Leslie  Ellis  (1863).^  Of  the  four  for- 
mulas for  oblique  spherical  triangles  which  are  sometimes  called  "Na- 
pier's Analogies,"  only  two  are  due  to  Napier  himself;  they  are  given 
in  his  Construdio.  The  other  two  were  added  by  Briggs  in  his  an- 
notations to  the  Construdio. 

A  modification  of  Napier's  logarithms  was  made  by  John  Speidell, 
a  teacher  of  mathematics  in  London,  who  pubHshed  the  New  Loga- 
rilhmes,  London,  1619,  containing  the  logarithms  of  sines,  tangents 
and  secants.  Speidell  did  not  advance  a  new  theory.  He  simply 
aimed  to  improve  on  Napier's  tables  by  making  all  logarithms  posi- 
tive. To  achieve  this  end  he  subtracted  Napier's  logarithmic  numbers 
from  10^  and  then  discarded  the  last  two  digits.  Napier  gave  log  sin 
3o'=474i38s2.  Subtracting  this  from  10^  leaves  52586148.  Speidell 
wrote  log  sin  30'  =  525861.  It  has  been  said  that  Speidell's  logarithms 
of  1619  are  logarithms  to  the  natural  base  e.  This  is  not  quitetrue, 
on  account  of  complications  arising  from  the  fact  that  the  logarithms 
in  Speidell's  table  appear  as  integral  numbers  and  that  the  natural 
trigonometric  values  (not  printed  in  Speidell's  tables)  are  likewise 
written  as  integral  numbers.  If  the  last  five  figures  in  Speidell's  log- 
arithms are  taken  as  decimals  (mantissas),  then  the  logarithms  are 
the  natural  logarithms  (with  10  added  to  every  negative  character- 
1  R.  Mortiz,  Am.  Math.  Monthly,  Vol.  22,  1915,  p.  221. 


VIETA  TO  DESCARTES  153 

istic)  of  the  trigonometric  values,  provided  the  latter  are  expressed 
decimally  as  ratios.  For  instance,  Napier  gives  sin  30' =  87 265,  the 
radius  being  10''.  In  reality,  sin  30' =  .0087 265.  The  natural  log- 
arithm of  this  fraction  is  approximately  5.25861.  Adding  10  gives 
5.25861.  As  seen  above,  Speidell  writes  log  sin  30'  =  525861.  The 
relation  between  the  natural  logarithms  and  the  logarithms  in  Spei- 
dell's  trigonometric  tables  is  shown  by  the  formula,  Sp.  log  jk;=io^ 

I  lo+loge  -^ ) .     For  secants  and  the  latter  haK  of  the  tangents  the 

addition  of  10  is  omitted.  In  Speidell's  table,  log  tan  89°=4048i2,  the 
natural  logarithm  of  tan  89°  being  4.04812.  In  the  1622  edition  of  his 
New  Logarithmes,  Speidell  included  also  a  table  of  logarithms  of  the 
numbers  i-iooo.  Except  for  the  omission  of  the  decimal  point,  the  loga- 
rithms in  this  table  are  genuinely  natural  logarithms.  Thus,  he  gives  log 
10=2302584;  in  modern  notation,  /o^eio=  2.302584.  J.  W.  L.  Glaisher 
has  pointed  out  ^  that  these  are  not  the  earliest  natural  logarithms.  The 
second  (1618)  edition  of  Edward  Wright's  translation  of  Napier's  De- 
scriptio  contains  an  anonymous  Appendix,  very  probably  written  by 
William  Oughtred,  describing  a  process  of  interpolation  with  the  aid 
of  a  small  table  containing  the  logarithms  of  72  sines.  The  latter 
are  natural  logarithms  with  the  decimal  point  omitted.  Thus,  log 
10=2302584,  log  50=3911021.  This  Appendix  is  noteworthy  also  as 
containing  the  earliest  account  of  the  radix  method  of  computing  log- 
arithms. After  the  time  of  Speidell  no  tables  of  natural  logarithms 
were  published  until  1770,  when  J.  H.  Lambert  inserted  a  seven  place 
table  of  natural  logarithms  of  the  numbers  i-ioo  in  his  Zusdlze  -,11  den 
Logarilhmischen  und  Trigonometrischen  Tabcllen.  Most  of  the  early 
methods  of  computing  logarithms  originated  in  England.  Napier 
begins  the  computations  of  his  logarithms  of  1614  by  forming  a  geo- 
metric progression  of  loi  terms,  the  first  term  being  10^  and  the  com- 
mon ratio ( I =  land  the  last  term  9,999,900.0004950.  This  progres- 
sion constitutes  the  "First  Table"  given  in  his  Constructio.  Omitting 
the  decimal  part  of  the  last  term,  he  takes  9,999,900  as  the  second 
term  of  a  new  progression  of  51  terms  whose  first  term  is  10^,  the  com- 

being  ( i -)  and  the  last  term  9,995,001.222927  (should 

be  9,995001.224804).  A  third  geometric  progression  of  21  terms  has 
10'  as  its  first  term,  9,995000  for  its  second  term,  the  common  ratio 

(I )  and  9,900,473.57808  as  its  last  term.    This  progression  ot 

\     2000/ 

zz  terms  constitutes  the  first  of  60  columns  of  numbers  in  Napier's 
-  \/uarierly  Jour,  of  Pure  &■  Appl.  Math.,  Vol.  40,  lois.  P   145. 


mon  ratio 


154  A  HISTORY  OF  MATHEMATICS 

"Third  Table."    Each  column  is  a  geometric  progression  of  21  terms 

with  I  I )  as  the  common  ratio.    The  60  first  or  top  numbers  in 

V      2000/ 

the  69  columns  themselves  constitute  a  geometric  progression  having 

the  ratio  ( i ) ,  the  first  top  number  being  10^,  the  second  9900000, 

and  so  on.  The  last  number  in  the  69th  column  is  4998609.4034. 
Thus  this  "Third  Table"  gives  a  series  of  numbers  very  nearly,  but 
not  exactly  in  geometrical  progression,  and  lying  between  10^  and 
very  nearly  l.io'^.  Says  Hutton,  these  tables  were  "found  in  the 
most  simple  manner,  by  Httle  more  than  easy  subtractions."  The 
numbers  are  taken  as  the  sines  of  angles  between  90°  and  30°.  Kine- 
matical  considerations  yield  him  an  upper  and  a  lower  limit  for  the 
logarithm  of  a  given  sine.  By  these  limits  he  obtains  the  logarithm 
of  each  number  in  his  "Third  Table."  To  obtain  the  logarithms  of 
sines  between  0°  and  30°  Napier  indicates  two  methods.  By  one  of 
them  he  computes  log  sin  ^,  i5°<  ^<3o°,  by  the  aid  of  his  "Third 
Table"  and  the  formula  sin  2^=2  sin^  sin(90°—  6).  A  repetition 
of  this  process  gives  the  logarithms  of  sines  down  to  ^  =  7°  30',  and 
so  on. 

Biirgi's  method  of  computation  was  more  primitive  than  Napier's. 
In  his  table  the  logarithms  were  printed  in  red  and  were  called  "red 
numbers " ;  the  antilogarithms  were  in  black.    The  expressions  r^.  =  ion, 

bn=bn—i   ( i-l 4),  where  ro=o,  ^(0=  100,000,000,  and  w=i,  2,3,..., 

indicate  the  mode  of  computation.      Any  term  ba  of  the  geometric 

series  is  obtained  by  adding  to  the  preceding  term  bn—i,  the  — 5th  part 

of  that  term.  Proceeding  thus  Biirgi  arrives  at  r=  230,270,022  and 
Z>=  1,000,000,000,  this  last  pair  of  numbers  being  obtained  by  inter- 
polation. 

In  the  Appendix  to  the  Constructio  there  are  described  three  meth- 
ods of  computing  logarithms  which  are  probably  the  result  of  the 
joint  labors  of  Napier  and  Briggs.  The  first  method  rests  on  the 
successive  extractions  of  fifth  roots.  The  second  calls  for  square 
roots  only.  Taking  log  1  =  0  and  log  io=io^°,  find  the  logarithm 
of  the  mean  proportion  between  i  and  10.  There  follows  log  v  i  x  10 
=log  3.16227766017  =  1  (io^°);  then  log  a/ 10x3. 162 2 776601 7  =  log 
5.62341325191  =  1  (10^"),  and  so  on.  Substantially  this  method  was 
used  by  Kepler  in  his  book  on  logarithms  of  1624  and  by  Vlacq.  The 
third  method  in  the  Appendix  to  the  Constructio  lets  log  1=0,  log  10= 
io^°,  and  takes  2  as  a  factor  io^°  times,  yielding  a  number  composed 
of  301029996  figures;  hence  log  2  =0,301029996. 


VIETA  TO  DESCARTES  15;; 

A  famous  method  of  computing  logarithms  is  the  so-called  "radix 
method."    It  requires  the  aid  of  a  table  of  radices  or  numbers  of  the 

T  ,  .  . 

form  1=1= — ,  with  their  logarithms.      The  logarithm  of  a  number  is 

f 

found  by  resolving  the  number  into  factors  of  the  form  i± —  and 

10° 

then  adding  the  logarithms  of  the  factors.  The  earliest  appearance 
of  this  method  is  in  the  anonymous  "Appendix"  (very  probably  due 
to  Oughtred)  to  Edward  Wright's  1618  edition  of  Napier's  Descriptio} 
It  is  fully  developed  by  Briggs  who,  in  his  Arithmetica  logarithniica, 
1624,  gives  a  table  of  radices.  The  method  has  been  frequently  re- 
discovered and  given  in  various  forms. ^  A  slight  simplification  of 
Briggs'  process  was  given  as  one  of  three  methods  by  Robert  Flower  in 
a  tract,  The  Radix  a  new  way  of  making  Logarithms,  London,  177 1. 
He  divides  a  given  number  by  a  power  of  10  and  a  single  digit,  so  as 
to  reduce  the  first  figure  to  .9,  and  then  multiplies  by  a  procession  of 
radices  until  all  the  digits  becomt  nines.  The  radix  method  was  re- 
discovered in  1786  by  George  Atwood  (i 746-1807),  the  inventor  of 
"Atwood's  machine,"  in  An  essay  on  the  Arithmetic  of  Factors,  and 
again  by  Zecchini  LeoneUi  in  1802,  by  Thomas  Manning  (1772-1S40), 
scholar  of  Caius  College,  Cambridge,  in  1806,  by  Thomas  Weddle  in 
1845,  Hearn  in  1847  and  Orchard  in  1848.  Extensions  and  variations 
of  the  radix  method  have  been  published  by  Peter  Gray  (i8o7?-iS87), 
a  writer  on  life  contingencies,  Thoman,  A.  J.  Ellis  (1814-1S90),  and 
others.  The  three  distinct  methods  of  its  application  are  due  to  Briggs, 
Flower  and  Weddle. 

Another  method  of  computing  common  logarithms  is  by  the  re- 
peated formation  of  geometric  means.  If  A  =  i,  B  =  io,  then  C= 
\/.45  =  3. 162278  has  the  logarithm  .5,  D=  V5C= 5.623413  has  the 
logarithm  .75,  etc.  Perhaps  suggested  by  Napier's  remarks  in  the 
Constructio,  this  method  was  developed  by  French  writers,  of  whom 
Jacques  Ozanam  (1640-1717)  in  1670  was  perhaps  the  first. ^  Ozanam 
is  best  known  for  his  Recreations  mathematiqties  et  physiques,  1694. 

Still  different  devices  for  the  computation  of  logarithms  were  in- 
vented by  Brook  Taylor  (1717),  John  Long  (1714),  William  Jones, 
Roger  Cotes  (1722),  Andrew  Reid  (1767),  James  Dodson  (1742),  Abel 
Biirja  (1786),  and  others.^ 

'  J.  W.  L.  Glaisher,  in  Quarterly  Jour,  of  Math's,  Vol.  46,  1915,  p.  125. 

^  For  the  detailed  history  of  this  method  consult  also  A.  J.  Ellis  in  Proceedings 
of  the  Royal  Society  (London),  Vol.  31,  1881,  pp.  398-413;  S.  Lupton,  Mathematical 
Gazette,  Vol.  7,  1913,  pp.  147-150,  170-173;  Ch.  Mutton's  Introduction  to  his 
Mathematical  Tables. 

^  See  J.  W.  L.  Glaisher  in  Quarterly  Journal  of  Pure  and  Appl.  Math's,  Vol.  47, 
1916,  pp.  249-301. 

•*  For  details  see  Ch.  Hutton's  Introduction  to  his  Mathematical  Tables,  also  the 
Encyclopedic  dcs  sciences  mathimatiques ,  1908;  I,  23,  "Tables  de  logarithmes.'" 


156  A  HISTORY  OF  MATHEMATICS 

After  the  labor  of  computing  logarithms  was  practically  over,  the 
facile  methods  of  computing  by  infinite  series  came  to  be  discovered. 
James  Gregory,  Lord  William  Brounker  (1620-1684),  Nicholas  Mer- 
cator  (1620-1687),  John  WalHs  and  Edmund  Halley  are  the  pioneer 
workers.  Mercator  in  1668  derived  what  amounts  to  the  infinite 
series  for  log  (i+a).  Transformations  of  this  series  yielded  rapidly 
converging  results.  WaUis  in  1695  obtained  |  log  (i+z)/log  (1-2)  = 
2-i-i  z^+l  z^+  .  ,  .  .  G.  Vega  in  his  Thesaurus  of  1794  lets  z=  i/(2y-—  i). 

The  theoretic  view  point  of  the  logarithm  was  broadened  somewhat 
during  the  seventeenth  century  by  the  graphic  representation,  both 
in  rectangular  and  polar  coordinates,  of  a  variable  and  its  variable 
logarithm.  Thus  were  invented  the  logarithmic  curve  and  the  loga- 
rithmic spiral.  It  has  been  thought  that  the  earliest_ reference  to  the 
logarithmic  curve  was  made  by  the  Italian  Evangelista  Torricelli  in 
a  letter  of  the  year  1644,  but  Paul  Tannery  made  it  practically  certain 
that  Descartes  knew  the  curve  in  1639.^  Descartes  described  the  log- 
arithmic spiral  in  1638  in  a  letter  to  P.  Mersenne,  but  does  not  give  its 
equation,  nor  connect  it  with  logarithms.  He  describes  it  as  the  curve 
which  makes  equal  angles  with  all  the  radii  drawn  through  the  origin. 
The  name  "logarithmic  spiral"  was  coined  by  Pierre  Varignon  in  a 
paper  presented  to  the  Paris  academy  in  1704  and  pubHshed  in  1722.^ 

The  most  brilliant  conquest  in  algebra  during  the  skteenth  century 
had  been  the  solution  of  cubic  and  biquadratic  equations.  All  at- 
tempts at  solving  algebraically  equations  of  higher  degrees  remaining 
fruitless,  a  new  line  of  inquiry — the  properties  of  equations  and  their 
roots— was  gradually  opened  up.  We  have  seen  that  Vieta  had  at- 
tained a  partial  knowledge  of  the  relations  between  roots  and  co- 
efficients. Jacques  Peletier  (15 17-158 2),  a  French  man  of  letters,  poet 
and  mathematician,  had  observed  as  early  as  1558,  that  the  root  of 
an  equation  is  a  divisor  of  the  last  term.  In  passing  he  writes  equa- 
tions with  all  terms  on  one  side,  and  equated  to  zero.  This  was  done 
also  by  Buteo  and  Harriot.  One  who  extended  the  theory  of  equa- 
tions somewhat  further  than  Vieta,  was  Albert  Girard  (1590?-! 633?), 
a  mathematician  of  Lorraine.  Like  Vieta,  this  ingenious  author  ap- 
plied algebra  to  geometry,  and  was  the  first  who  understood  the  use 
of  negative  roots  in  the  solution  of  geometric  problems.  He  spoke  of 
imaginary  quantities,  inferred  by  induction  that  every  equation  has 
as  many  roots  as  there  are  units  in  the  number  expressing  its  degree, 
and  first  showed  how  to  express  the  sums  of  their  powers  in  terms  of 
the  coefficients.  Another  algebraist  of  considerable  power  was  the 
English  Thomas  Harriot  (i 560-1621).  He  accompanied  the  first 
colony  sent  out  by  Sir  Walter  Raleigh  to  Virginia.    After  having  sur- 

iSee  G.  Loria,  Bihliotheca  math.,  3.  S.,  Vol.  i,  1900,  p.  75;  Uintermediaire  des 
matheniaticiens,  Vol.  7,  1900,  p.  95.  .  ,        j 

2  For  details  and  references,  see  F.  Cajori,  "History  of  the  Exponential  and 
Logarithmic  Concepts,"  Am.  Math.  Monthly,  Vol.  20,  1913,  pp.  10,  11. 


VIETA  TO  DESCARTES  157 

veyed  that  country  he  returned  to  England.  As  a  mathematician,  he 
was  the  boast  of  his  country.  He  brought  the  theory  of  equations 
under  one  comprehensive  point  of  view  by  grasping  that  truth  in  its 
full  extent  to  which  Vieta  and  Girard  only  approximated;  viz.  that 
in  an  equation  in  its  simplest  form,  the  coefficient  of  the  second  term 
with  its  sign  changed  is  equal  to  the  sum  of  the  roots;  the  coefficient 
of  the  third  is  equal  to  the  sum  of  the  products  of  every  two  of  the 
roots,  etc.  He  was  the  first  to  decompose  equations  into  their  simple 
factors;  but,  since  he  failed  to  recognize  imaginary  and  even  negative 
roots,  he  failed  also  to  prove  that  every  equation  could  be  thus  de- 
composed. Harriot  made  some  changes  in  algebraic  notation,  adopt- 
ing small  letters  of  the  alphabet  in  place  of  the  capitals  used  by  Vieta. 
The  symbols  of  inequality  >  and  <  were  introduced  by  him.  The 
signs  ^  and  S  were  first  used  about  a  century  later  by  the  Parisian 
hydrographer,  Pierre  Bouguer.^  Harriot's  work,  A rtis  A  nalyticce  praxis, 
was  published  in  1631,  ten  years  after  his  death.  William  Oughtred 
(1574-1660)  contributed  vastly  to  the  propagation  of  mathematical 
knowledge  in  England  by  his  treatises,  the  Clavis  maihematica,  163 1 
(later  Latin  editions,  1648,  1652,  1667,  1693;  EngHsh  editions,  1647, 
1694),  Circles  of  Proportion,  1632,  Trigonomctrie,  1657.^  Oughtred 
was  an  episcopal  minister  at  Albury,  near  London,  and  gave  private 
lessons,  free  of  charge,  to  pupils  interested  in  mathematics.  Among 
his  most  noted  pupils  are  the  mathematician  John  Wallis  and  the 
astronomer  Seth  Ward.  Oughtred  laid  extraordinary  emphasis  upon 
the  use  of  mathematical  symbols;  altogether  he  used  over  150  of  them. 
Only  three  have  come  down  to  modern  times,  namely  X  as  the  symbol 
of  multiplication,  ::  as  that  of  proportion,  and  ^-^  as  that  for  "differ- 
ence." The  symbol  X  occurs  in  the  Clavis,  but  the  letter  X  which 
closely  resembles  it,  occurs  as  a  sign  of  multiplication  in  the  anony- 
mous "Appendix  to  the  Logarithmes"  in  Edward  Wright's  transla- 
tion of  Napier's  Dcscriptio,  pubhshcd  in  1618.^  This  appendix  was 
most  probably  written  by  Oughtred.  A  proportion  A:B  =  C:D  he 
wrote  A-  B::  C-  D.  Oughtred's  notation  for  ratio  and  proportion  was 
widely  used  in  England  and  on  the  Continent,  but  as  early  as  165 1 
the  English  astronomer  Vincent  Wing  began  to  use  (:)  for  ratio,''  a 
notation  which  gained  ground  and  freed  the  dot  (.)  for  use  as  the  sym- 
bol of  separation  in  decimal  fractions.  It  is  interesting  to  note  the 
attitude  of  Leibniz  toward  some  of  these  symbols.  On  July  29,  1698, 
he  wrote  in  a  letter  to  John  Bernoulli:  "I  do  not  like  X  as  a  symb')l 
for  multiplication,  as  it  is  easily  confounded  with  .t;  .  .  .  often  I  simply 
relate  two  quantities  by  an  interposed  dot  and  indicate  multiplication 

'  P.  H.  Fuss,  Corresp.  math,  phys.,  I,  1843,  P-  304;  Eiicydopedie  des  sciences  maitii 
maliques,  T.  I,  VoL  I,  1904,  p.  23. 

*See  F.  Cajori,  William  Oughtred,  Chicago  and  London,  1916. 
5  F.  Cajcri,  in  Nature,  Vol.  XCIV,  19 14,  p.  363. 
*  Ibid.,  p.  477. 


X58  A  HISTORY  OF  MATHEMATICS 

by  ZC-LM.  Hence  in  designating  ratio  I  use  not  one  point  but  two 
points,  which  I  use,  at  the  same  time,  for  division;  thus,  for  your 
dy.xy.dt.a  I  write  dy:x=dt: a;  for,  y  is  to  x  as  dt  is  to  a,  is  indeed  the 
same  as,  dy  divided  by  x  is  equal  to  dt  divided  by  a.  From  this  equa- 
tion follow  then  all  the  rules  of  proportion."  This  conception  of 
ratio  and  proportion  was  far  in  advance  of  that  in  contemporary 
arithmetics.  Through  the  aid  of  Christian  Wolf  the  dot  was  generally 
adopted  in  the  eighteenth  century  as  a  symbol  of  multiplication. 
Presumably  Leibniz  had  no  knowledge  that  Harriot  in  his  Artis 
analvticcs  praxis,  1631,  used  a  dot  for  multiplication,  as  in  aaa—T,- 
bba  =  +  2.cc:c.  Harriot's  clot  received  no  attention,  not  even  from  WaUis. 
Oughtred  and  some  of  his  English  contemporaries,  Richard  Nor- 
wood, John  Speidell  and  others  were  prominent  in  introducing  abbre- 
viations for  the  trigonometric  functions:  s,  si,  or  sin  for  sine;  s  co  or 
si  CO  for  "sine  complement"  or  cosine;  se  for  secant,  etc.  Oughtred 
did  not  use  parentheses.  Terms  to  be  aggregated  were  enclosed  be- 
tween double  colons.  He  wrote  V(yl+£)  thus,  ylq:A+E:  The  two 
dots  at  the  end  were  sometimes  omitted.  Thus,  C:A+B—E  meant 
(A+B  —  E).^  Before  Oughtred  the  use  of  parentheses  had  been  sug- 
gested by  Clavius  in  1608  and  Girard  in  1629.    In  fact,  as  early  as 

1556  Tartaglia  wrote  \/^28  — Vio  thus  R  v.  (R28  men  Rio),  where 
R  V.  means  "radix  universahs,"  but  he  did  not  use  parentheses  in  in- 
dicating the  product  of  two  expressions.^  Parentheses  were  used  by 
I.  Errard  de  Bar-le-Duc  (1619),  Jacobo  de  Billy  (1643),  Richard 
Norwood  (1631),  Samuel  Foster  (1659);  nevertheless  parentheses  did 
not  become  popular  in  algebra  before  the  time  of  Leibniz  and  the 
Bernoullis. 

It  is  noteworthy  that  Oughtred  denotes  Si  and  f  f|,  the  approxi- 
mate ratios  of  the  circumference  to  the  diameter,  by  the  S3Tnbol  ^;  it 

occurs  in  the  1647  edition  and  in  the  later  editions  of  his  Clavis  mathe- 
maticcB.  Oughtred's  notation  was  adopted  and  used  extensively  by 
Isaac  Barrow.  It  was  the  forerunner  of  the  notation  7r=3.i4i59  .  .  .  , 
first  used  by  William  Jones  in  1706  in  his  Synopsis  palmanorum  ma- 
thcseos,  London,  1706,  p.  263.  L.  Euler  first  used  7r=3.i4i59  •  •  •  in 
1737.    In  his  time,  the  symbol  met  with  general  adoption. 

Oughtred  stands  out  prominently  as  the  inventor  of  the  circular 
and  the  rectilinear  slide  rules.  The  circular  shde  rule  was  described 
in  print  in  his  book,  the  Circles  of  Proportion,  1632.  His  rectilinear 
slide  rule  was  described  in  1633  in  an  Addition  to  the  above  work. 
But  Oughtred  was  not  the  first  to  describe  the  circular  sUde  rule  in 
print;  this  was  done  by  one  of  his  pupils,  Richard  Delamain,  in  1630, 
in  a  booklet,  entitled  Gratnmelogia?    A  bitter  controversy  arose  be- 

1  G.  Enestrom  in  Bibliotheca  mathematica,  3.  S.,  Vol.  7,  p.  296. 

2  See  F.  Cajori,  William  Oiighlred,  Chicago  and  London,  1916,  p.  46. 


VIETA  TO  DESCARTES  159 

tween  Delamain  and  Oughtred.  Each  accused  the  other  of  having 
stolen  the  invention  from  him.  Most  probably  each  was  an  in- 
dependent inventor.  To  the  invention  of  the  rectihnear  slide  rule 
Oughtred  has  a  clear  title.  He  states  that  he  designed  his  shde  rules 
as  early  as  1621.  The  slide  rule  was  improved  in  England  during  the 
seventeenth  and  eighteenth  centuries  and  was  used  quite  extensively.^ 

Some  of  the  stories  told  about  Oughtred  are  doubtless  apocryphal, 
as  for  instance,  that  his  economical  wife  denied  him  the  use  of  a  candle 
for  study  in  the  evening,  and  that  he  died  of  joy  at  the  Restoration, 
after  drinking  "a  glass  of  sack"  to  his  Majesty's  health.  De  Morgan 
humorously  remarks,  "It  should  be  added,  by  way  of  excuse,  that  he 
was  eighty-six  years  old." 

Algebra  was  now  in  a  state  of  sufficient  perfection  to  enable  Des- 
cartes and  others  to  take  that  important  step  which  forms  one  of  the 
grand  epochs  in  the  history  of  mathematics, — the  application  of  alge- 
braic analysis  to  define  the  nature  and  investigate  the  properties  of 
algebraic  curves. 

In  geometry,  the  determination  of  the  areas  of  curvilinear  figures 
was  diligently  studied  at  this  period.  Paul  Guldin  (1577-1643),  a 
Swiss  mathematician  of  considerable  note,  rediscovered  the  following 
theorem,  published  in  his  Centroharyca,  which  has  been  named  after 
him,  though  first  found  in  the  Mathematical  Collections  of  Pappus: 
The  volume  of  a  solid  of  revolution  is  equal  to  the  area  of  the  generat- 
ing figure,  multiplied  by  the  circumference  described  by  the  centre  of 
gravity.  We  shall  see  that  this  method  excels  that  of  Kepler  and 
Cavalieri  in  following  a  more  exact  and  natural  course;  but  it  has  the 
disadvantage  of  necessitating  the  determination  of  the  centre  of  grav- 
ity, which  in  itself  may  be  a  more  difficult  problem  than  the  original 
one  of  finding  the  volume.  Guldin  made  some  attempts  to  prove  his 
theorem,  but  Cavalieri  pointed  out,  the  weakness  of  his  demonstration. 

Johannes  Kepler  (1571-1630)  was  a  native  of  Wiirtemberg  and  im- 
bibed Copernican  principles  while  at  the  University  of  Tubingen.  His 
pursuit  of  science  was  repeatedly  interrupted  by  war,  religious  perse- 
cution, pecuniary  embarrassments,  frequent  changes  of  residence, 
and  family  troubles.  In  1600  he  became  for  one  year  assistant  to  the 
Danish  astronomer,  Tycho  Brahe,  in  the  observatory  near  Prague. 
The  relation  between  the  two  great  astronomers  was  not  always  of  an 
agreeable  character.  Kepler's  publications  are  voluminous.  His  first  at- 
tempt to  explain  the  solar  system  was  made  in  1596,  when  he  thought 
he  had  discovered  a  curious  relation  between  the  five  regular  solids 
and  the  number  and  distance  of  the  planets.  The  pubhcation  of  this 
pseudo-discovery  brought  him  much  fame.  At  one  time  he  tried  to 
represent  the  orbit  of  Mars  by  the  oval  curve  which  we  now  write  in 
polar  coordinates,  p  =  2r  cos^  ^.  Maturer  reflection  and  intercourse 
with  Tycho  Brahe  and  Galileo  led  him  to  investigations  and  results 
1  See  F.  Cajori,  History  of  the  Logarithmic  Slide  Rule,  New  York,  1909. 


lOo  A  HISTORY  OF  MATHEMATICS 

worthy  of  his  genius — "Kepler's  laws."  He  enriched  pure  mathe- 
matics as  well  as  astronomy.  It  is  not  strange  that  he  was  interested 
in  the  mathematical  science  which  had  done  him  so  much  service;  for 
"if  the  Greeks  had  not  cultivated  conic  sections,  Kepler  could  not 
have  superseded  Ptolemy."  ^  The  Greeks  never  dreamed  that  these 
curves  would  ever  be  of  practical  use;  Aristasus  and  ApoUonius 
studied  them  merely  to  satisfy  their  intellectual  cravings  after  the 
ideal;  yet  the  conic  sections  assisted  Kepler  in  tracing  the  march  of 
the  planets  in  their  elliptic  orbits.  Kepler  made  also  extended  use  of 
logarithms  and  decimal  fractions,  and  was  enthusiastic  in  diffusing 
a  knowledge  of  them.  At  one  time,  while  purchasing  wine,  he  was 
struck  by  the  inaccuracy  of  the  ordinary  modes  of  determining  the 
contents  of  kegs.  This  led  him  to  the  study  of  the  volumes  of  solids 
of  revolution  and  to  the  publication  of  the  Stereometria  Doliorum  in 
1615.  In  it  he  deals  first  with  the  solids  known  to  Archimedes  and 
then  takes  up  others.  Kepler  made  wide  appHcation  of  an  old  but 
neglected  idea,  that  of  infinitely  great  and  infinitely  small  quantities. 
Greek  mathematicians  usually  shunned  this  notion,  but  with  it  modern 
mathematicians  completely  revolutionized  the  science.  In  comparing 
rectilinear  figures,  the  method  of  superposition  was  employed  by  the 
ancients,  but  in  comparing  rectilinear  and  curvilinear  figures  with 
each  other,  this  method  failed  because  no  addition  or  subtraction  of 
rectilinear  figures  could  ever  produce  curvilinear  ones.  To  meet  this 
case,  they  devised  the  Method  of  Exhaustion,  which  was  long  and 
difficult;  it  was  purely  synthetical,  and  in  general  required  that  the 
conclusion  should  be  known  at  the  outset.  The  new  notion  of  infinity 
led  gradually  to  the  invention  of  methods  immeasurably  more  power- 
ful. Kepler  conceived  the  circle  to  be  composed  of  an  infinite  number 
of  triangles  having  their  common  vertices  at  the  centre,  and  their 
bases  in  the  circumference;  and  the  sphere  to  consist  of  an  infinite 
number  of  pyramids.  He  applied  conceptions  of  this  kind  to  the  de- 
termination of  the  areas  and  volumes  of  figures  generated  by  curves 
revolving  about  any  line  as  axis,  but  succeeded  in  solving  only  a  few 
of  the  simplest  out  of  the  84  problems  which  he  proposed  for  investi- 
gation in  his  Stereometria. 

Other  points  of  mathematical  interest  in  Kepler's  works  are  (i)  the 
assertion  that  the  circumference  of  an  ellipse,  whose  axes  are  2a  and 
2h,  is  nearly  tv  {a-\-b);  (2)  a  passage  from  which  it  has  been  inferred 
that  Kepler  knew  the  variation  of  a  function  near  its  maximum  value 
to  disappear;  (3)  the  assumption  of  the  principle  of  continuity  (which 
differentiates  modern  from  ancient  geometry),  when  he  shows  that 
a  parabola  has  a  focus  at  infinity,  that  lines  radiating  from  this  "  caecus 
focus"  are  parallel  and  have  no  other  point  at  infinity. 

The  Stereometria  led  Cavalieri,  an  Italian  Jesuit,  to  the  consideration 

1  William  Whewell,  History  of  the  Inductive  Sciences,  3rd  Ed.,  New  York,  1858, 
Vol.  I,  p.  311. 


VIETA  TO  DESCARTES  i6i 

of  infinitely  small  quantities.  Bonaventura  Cavalieri  (i 598-1647), 
a  pupil  of  Galileo  and  professor  at  Bologna,  is  celebrated  for  his  Geo- 
metria  imlivisibiUhis  contimiorum  nova  qiiadmn  ralione  promota,  1635. 
This  work  expounds  his  method  of  Indivisibles,  which  occupies  an 
intermediate  place  between  the  method  of  exhaustion  of  the  Greeks 
and  the  methods  of  Newton  and  Leibniz.  "Indivisibles"  were  dis- 
cussed by  Aristotle  and  the  scholastic  philosophers.  They  commanded 
the  attention  of  Galileo.  Cavalieri  does  not  define  the  term.  He 
borrows  the  concept  from  the  scholastic  philosophy  of  Bradwardine 
and  Thomas  Aquinas,  in  which  a  point  is  the  indivisible  of  a  line,  a  line 
the  indivisible  of  a  surface,  etc.  Each  indivisible  is  capable  of  gener- 
ating the  next  higher  continuum  by  motion;  a  moving  point  generates 
a  line,  etc.  The  relative  magnitude  of  two  solids  or  surfaces  could 
then  be  found  simply  by  the  summation  of  series  of  planes  or  lines. 
For  example,  Cavalieri  finds  the  sum  of  the  squares  of  all  lines  making 
up  a  triangle  equal  to  one-third  the  sum  of  the  squares  of  all  lines  of 
a  parallelogram  of  equal  base  and  altitude;  for  if  in  a  triangle,  the  first 
line  at  the  apex  be  i,  then  the  second  is  2,  the  third  is  3,  and  so  on; 
and  the  sum  of  their  squares  is 

12+22-H32+  .  .  .  -\-n'^=n{n+i)  (2w+i)-^6. 
In  the  parallelogram,  each  of  the  lines  is  n  and  their  number  is  n;  hence 
the  total  sum  of  their  squares  is  n^.    The  ratio  between  the  two  sums 
is  therefore 

n{n+i)  {2n+i)-^()n^=\, 

since  n  is  infinite.  From  this  he  concludes  that  the  pyramid  or  cone  is 
respectively  |  of  a  prism  or  cylinder  of  equal  base  and  altitude,  since 
the  polygons  or  circles  composing  the  former  decrease  from  the  base 
to  the  apex  in  the  same  way  as  the  squares  of  the  lines  parallel  to  the 
base  in  a  triangle  decrease  from  base  to  apex.  By  the  ^lethod  of  In- 
divisibles, Cavalieri  solved  the  majority  of  the  problems  proposed  by 
Kepler,  Though  expeditious  and  yielding  correct  results,  Cavalieri's 
method  lacks  a  scientific  foundation.  If  a  line  has  absolutely  no  width, 
then  the  addition  of  no  number,  however  great,  of  lines  can  ever  yield 
an  area;  if  a  plane  has  no  thickness  whatever,  then  even  an  infinite 
number  of  planes  cannot  form  a  solid.  Though  unphilosophicc: 
Cavalieri's  method  was  used  for  fifty  years  as  a  sort  of  integral 
calculus.  It  yielded  solutions  to  some  difficult  problems.  Guldin 
made  a  severe  attack  on  Cavalieri  and  his  method.  The  latter 
published  in  1647,  after  the  death  of  Guldin,  a  treatise  entitled 
Exercitaiiones  geometric^  sex,  in  which  he  replied  to  the  objections 
of  his  opponent  and  attempted  to  give  a  clearer  explanation  of  his 
method.  Guldin  had  never  been  able  to  demonstrate  the  theorem 
named  after  him,  except  by  metaphysical  reasoning,  but  Cavalieri 
proved  it  by  the  method  of  indivisibles.  A  revised  edition  of  the 
Ccometria  appeared  in  1653. 


i62  A  HISTORY  OF  MATHEMATICS 

There  is  an  important  curve,  not  known  to  the  ancients,  which  now 
began  to  be  studied  with  great  zeal.  Roberval  gave  it  the  name  of 
"trochoid,"  Pascal  the  name  of  "roulette,"  Galileo  the  name  of  "cy- 
cloid." The  invention  of  this  curve  seems  to  be  due  to  Charles  Bou- 
velles  who  in  a  geometry  published  in  Paris  in  1501  refers  to  this  curve 
in  connection  with  the  problem  of  the  squaring  of  the  circle.  Galileo 
valued  it  for  the  graceful  form  it  would  give  to  arches  in  architecture. 
He  ascertained  its  area  by  weighing  paper  figures  of  the  cycloid  against 
that  of  the  generating  circle,  and  found  thereby  the  first  area  to  be 
nearly  but  not  exactly  thrice  the  latter.  A  mathematical  determina- 
tion was  made  by  his  pupil,  Evangelista  Torricelli  (1608-1647),  who 
is  more  widely  known  as  a  physicist  than  as  a  mathematician. 

By  the  Method  of  Indivisibles  he  demonstrated  its  area  to  be  triple 
that  of  the  revolving  circle,  and  pubUshed  his  solution.  This  same 
quadrature  had  been  effected  a  few  years  earlier  (about  1636)  by 
Roberval  in  France,  but  his  solution  was  not  known  to  the  Italians. 
Roberval,  being  a  man  of  irritable  and  violent  disposition,  unjustly 
accused  the  mild  and  amiable  Torricelli  of  stealing  the  proof.  This 
accusation  of  plagiarism  created  so  much  chagrin  with  Torricelli  that 
it  is  considered  to  have  been  the  cause  of  his  early  death.  Vincenzo 
Viviani  (1622-1703),  another  prominent  pupil  of  Galileo,  determined 
the  tangent  to  the  cycloid.  This  was  accomplished  in  France  by 
Descartes  and  Fermat. 

In  France,  where  geometry  began  to  be  cultivated  with  greatest 
success,  Roberval,  Fermat,  Pascal,  employed  the  Alethod  of  Indivis- 
ibles and  made  new  improvements  in  it.  Giles  Persone  de  Roberval 
(1602-1675),  for  forty  years  professor  of  mathematics  at  the  College 
of  France  in  Paris,  claimed  for  himself  the  invention  of  the  Method  of 
Indivisibles.  Since  his  complete  works  were  not  published  until  after 
his  death,  it  is  difficult  to  settle  questions  of  priority.  Montucla  and 
Chasles  are  of  the  opinion  that  he  invented  the  method  independently 
of  and  earlier  than  the  Italian  geometer,  though  the  work  of  the  latter 
was  pubUshed  much  earlier  than  Roberval's.  Marie  finds  it  difficult 
to  believe  that  the  Frenchman  borrowed  nothing  whatever  from  the 
Italian,  for  both  could  not  have  hit  independently  upon  the  word 
Indivisibles,  which  is  applicable  to  infinitely  small  quantities,  as  con- 
ceived by  Cavalieri,  but  not  as  conceived  by  Roberval.  Roberval 
and  Pascal  improved  the  rational  basis  of  the  Method  of  Indivisibles, 
by  considering  an  area  as  made  up  of  an  indefinite  number  of  rectangles 
instead  of  lines,  and  a  solid  as  composed  of  indefinitely  small  soHds 
instead  of  surfaces.  Roberval  applied  the  method  to  the  finding  of 
areas,  volumes,  and  centres  of  gravity.  He  effected  the  quadrature 
of  a  parabola  of  any  degree  y^=d^~^x,  and  also  of  a  parabola  y"  = 
Qin— n_^n^  Wc  havc  already  mentioned  his  quadrature  of  the  cycloid. 
Roberval  is  best  known  for  his  method  of  drawing  tangents,  '■vhich, 
however,  was  invented  at  the  same  time,  if  not  earlier,  by  Torricelli. 


VIETA  TO  DESCARTES  163 

Torricelli's  appeared  in  1644  under  the  title  Opera  geomelrica.  Rober- 
val  gives  the  fuller  exposition  of  it.  Some  of  his  special  applications 
were  published  at  Paris  as  early  as  1644  in  Mersenne's  Cogitala  physico- 
mathematica.  Roberval  presented  the  full  development  of  the  sub- 
ject to  the  French  Academy  of  Sciences  in  1668  which  published  it 
in  its  Memoires.  This  academy  had  grown  out  of  scientific  meetings 
held  with  Mersenne  at  Paris.  It  was  founded  by  JNIinister  Richelieu 
in  163s  and  reorganized  by  Minister  Colbert  in  1666.  Marin  Mersenne 
(15S8-1648)  rendered  great  services  to  science.  His  polite  and  en- 
gaging manners  procured  him  many  friends,  including  Descartes  and 
Fermat.  He  encouraged  scientific  research,  carried  on  an  extensive 
correspondence,  and  thereby  was  the  medium  for  the  intercommunica- 
tion of  scientific  intelligence. 

Roberval's  method  of  drawing  tangents  is  allied  to  Newton's  prin- 
ciple of  fluxions.  Archimedes  conceived  his  spiral  to  be  generated  by 
a  double  motion.  This  idea  Roberval  extended  to  all  curves.  Plane 
curves,  as  for  instance  the  conic  sections,  may  be  generated  by  a  point 
acted  upon  by  two  forces,  and  are  the  resultant  of  two  motions.  If 
at  any  point  of  the  curve  the  resultant  be  resolved  into  its  components, 
then  the  diagonal  of  the  parallelogram  determined  by  them  is  the  tan- 
gent to  the  curve  at  that  point.  The  greatest  difficulty  connected 
with  this  ingenious  method  consisted  in  resolving  the  resultant  into 
components  having  the  proper  lengths  and  directions.  Roberval  did 
not  always  succeed  in  doing  this,  yet  his  new  idea  was  a  great  step  in 
advance.  He  broke  off  from  the  ancient  definition  of  a  tangent  as 
a  straight  line  having  only  one  point  in  common  wdth  a  curve, — a  defi- 
nition which  by  the  methods  then  available  was  not  adapted  to  bring 
out  the  properties  of  tangents  to  curves  of  higher  degrees,  nor  even  of 
curv^es  of  the  second  degree  and  the  parts  they  may  be  made  to  play 
in  the  generation  of  the  cur^-es.  The  subject  of  tangents  received 
special  attention  also  from  Fermat,  Descartes,  and  Barrow,  and 
reached  its  highest  development  after  the  invention  of  the  differential 
calculus.  Fermat  and  Descartes  defined  tangents  as  secants  whose 
two  points  of  intersection  with  the  curve  coincide;  Barrow  considered 
a  curve  a  polygon,  and  called  one  of  its  sides  produced  a  tangent. 

A  profound  scholar  in  all  branches  of  learning  and  a  mathematician 
of  exceptional  powers  was  Pierre  de  Fermat  (1601-1665).  He  studied 
law  at  Toulouse,  and  in  163 1  was  made  councillor  for  the  parliament 
of  Toulouse.  His  leisure  time  was  mostly  devoted  to  mathematics, 
which  he  studied  with  irresistible  passion.  Unlike  Descartes  and 
Pascal,  he  led  a  quiet  and  unaggressive  life.  Fermat  has  left  the  im- 
press of  his  genius  upon  all  branches  of  mathematics  then  known. 
A  great  contribution  to  geometry  was  his  De  maximis  et  minimis. 
About  twenty  years  earlier,  Kepler  had  first  ol)served  that  the  incre- 
ment of  a  variable,  as,  for  instance,  the  ordinate  of  a  curve,  is  evan- 
escent for  values  very  near  a  maximum  or  a  minimu^n  value  of  the 


i64  A  HISTORY  OP  MATHEMATICS 

variable.  Developing  this  idea,  Fermat  obtained  his  rule  for  maxima 
and  minima.  He  substituted  x^-e  for  x  in  the  given  function  of  x  and 
then  equated  to  each  other  the  two  consecutive  values  of  the  function 
and  divided  the  equation  by  e.  If  e  be  taken  o,  then  the  roots  of  this 
equation  are  the  values  of  x,  making  the  function  a  maximum  or  a 
minimum.  Fermat  was  in  possession  of  this  rule  in  1629.  The  main 
difference  between  it  and  the  rule  of  the  differential  calculus  is  that  it 
introduces  the  indefinite  quantity  e  instead  of  the  infinitely  small  dx. 
Fermat  made  it  the  basis  for  his  method  of  drawing  tangents,  which 
involved  the  determination  of  the  length  of  the  subtangent  for  a  given 
point  of  a  curve. 

Owing  to  a  want  of  explicitness  in  statement,  Fermat's  method  of 
maxima  and  minima,  and  of  tangents,  was  severely  attacked  by  his 
great  contemporary,  Descartes,  who  could  never  be  brought  to  render 
due  justice  to  his  merit.  In  the  ensuing  dispute,  Fermat  found  two 
zealous  defenders  in  Roberval  and  Pascal,  the  father;  while  C.  My- 
dorge,  G.  Desargues,  and  Claude  Hardy  supported  Descartes. 

Since  Fermat  introduced  the  conception  of  iniinitely  small  differ- 
ences between  consecutive  values  of  a  function  and  arrived  at  the 
principle  for  finding  the  maxima  and  minima,  it  was  maintained  by 
Lagrange,  Laplace,  and  Fourier,  that  Fermat  may  be  regarded  as  the 
first  inventor  of  the  differential  calculus.  This  point  is  not  well  taken, 
as  will  be  seen  from  the  words  of  Poisson,  himself  a  Frenchman,  who 
rightly  says  that  the  differential  calculus  "consists  in  a  system  of  rules 
proper  for  finding  the  differentials  of  all  functions,  rather  than  in  the 
use  which  may  be  made  of  these  infinitely  small  variations  in  the  so- 
lution of  one  or  two  isolated  problems.") 

A  contemporary  mathematician,  whose  genius  perhaps  equalled  that 
of  the  great  Fermat,  was  Blaise  Pascal  (1623-1662).  He  was  born  at 
Clermont  in  Auvergne.  In  1026  his  father  retired  to  Paris,  where  he 
devoted  himself  to  teaching  his  son,  for  he  would  not  trusthis  educa- 
tion to  others.  Blaise  Pascal's  genius  for  geometry  showed  itself  when 
he  was  but  twelve  years  old.  His  father  was  well  skilled  in  mathe- 
matics, but  did  not  wish  his  son  to  study  it  until  he  was  perfectly 
acquainted  with  Latin  and  Greek.  All  mathematical  books  were 
hidden  out  of  his  sight.  The  boy  once  asked  his  father  what  mathe- 
matics treated  of,  and  was  answered,  in  general,  "that  it  was  the 
method  of  making  figures  with  exactness,  and  of  finding  out  what 
proportions  they  relatively  had  to  one  another."  He  was  at  the  same 
time  forbidden  to  talk  any  more  about  it,  or  ever  to  think  of  it.  But 
his  genius  could  not  submit  to  be  confined  within  these  bounds.  Start- 
ing with  the  bare  fact  that  mathematics  taught  the  means  of  making 
figures  infallibly  exact,  he  employed  his  thoughts  about  it  and  with 
a  piece  of  charcoal  drew  figures  upon  the  tiles  of  the  pavement,  trying 
the  methods  of  drawing,  for  example,  an  exact  circle  or  equilateral 
triangle.    He  gave  names  of  his  own  to  these  figures  and  then  formed 


VIETA  TO  DESCARTES  165 

axioms,  and,  in  short,  came  to  make  demonstrations.  In  this  way  he 
is  reported  to  have  arrived  unaided  at  the  theorem  that  the  sum  of 
the  three  angles  of  a  triangle  is  equal  to  two  right  angles.  His  father 
caught  him  in  the  act  of  studying  this  theorem,  and  was  so  astonished 
at  the  sublimity  and  force  of  his  genius  as  to  weep  for  joy.  The  father 
now  gave  him  Euclid's  Elements,  which  he,  without  assistance,  mas- 
tered easily.  His  regular  studies  being  languages,  the  boy  employed 
only  his  hours  of  amusement  on  the  study  of  geometry,  yet  he  had  so 
ready  and  lively  a  penetration  that,  at  the  age  of  sixteen,  he  wrote 
a  treatise  upon  conies,  which  passed  for  such  a  surprising  effort  of 
genius,  that  it  was  said  nothing  equal  to  it  in  strength  had  been  pro- 
duced since  the  time  of  Archimedes.  Descartes  refused  to  believe 
that  it  was  written  by  one  so  young  as  Pascal.  This  treatise  was  never 
published,  and  is  now  lost.  Leibniz  saw  it  in  Paris  and  reported  on 
a  portion  of  its  contents.  The  precocious  youth  made  vast  progress 
in  all  the  sciences,  but  the  constant  application  at  so  tender  an  age 
greatly  impaired  his  health.  Yet  he  continued  working,  and  at  nine- 
teen invented  his  famous  machine  for  performing  arithmetical  opera- 
tions mechanically.  This  continued  strain  from  overwork  resulted  in 
a  permanent  indisposition,  and  he  would  sometimes  say  that  from  the 
time  he  was  eighteen,  he  never  passed  a  day  free  from  pain.  At  the 
age  of  twenty-four  he  resolved  to  lay  aside  the  study  of  the  human 
sciences  and  to  consecrate  his  talents  to  religion.  His  Provincial 
Letters  against  the  Jesuits  are  celebrated.  But  at  times  he  returned 
to  the  favorite  study  of  his  youth.  Being  kept  awake  one  night  by 
a  toothache,  some  thoughts  undesignedly  came  into  his  head  concern- 
ing the  roulette  or  cycloid;  one  idea  followed  another;  and  he  thus 
discovered  properties  of  this  curve  even  to  demonstration.  A  corre- 
spondence between  him  and  Fermat  on  certain  problems  was  the 
beginning  of  the  theory  of  probabiUty.  Pascal's  illness  increased,  and 
he  died  at  Paris  at  the  early  age  of  thirty-nine  years.  By  him  the 
answer  to  the  objection  to  Cavalieri's  JMethod  of  Indivisibles  was  put 
in  clearer  form.  Like  Roberval,  he  explained  "  the  sum  of  right  lines  " 
to  mean  "the  sum  of  infinitely  small  rectangles."  Pascal  greatly  ad- 
vanced the  knowledge  of  the  cycloid.  He  determined  the  area  of  a 
section  produced  by  any  line  parallel  to  the  base;  the  volume  gener- 
ated by  it  revolving  around  its  base  or  around  the  axis;  and,  finally, 
the  centres  of  gravity  of  these  volumes,  and  also  of  half  these  volumes 
cut  by  planes  of  symmetry.  Before  publishing  his  results,  he  sent, 
in  1658,  to  all  mathematicians  that  famous  challenge  offering  prizes 
for  the  first  two  solutions  of  these  problems.  Only  Wallis  and  A.  La 
Louvere  competed  for  them.  The  latter  was  quite  unequal  to  the  task ; 
the  former,  being  pressed  for  time,  made  numerous  mistakes:  neither 
got  a  prize.  Pascal  then  published  his  own  solutions,  which  produced 
a  great  sensation  among  scientific  men.  Wallis,  too,  published  his. 
with  the  errors  corrected.     Though  not  competing  for  the  prizes. 


x66  A  HISTORY  OF  MATHEMATICS 

Huygens,  Wren,  and  Fermat  solved  some  of  the  questions.  The  chief 
discoveries  of  Christopher  Wren  (1632-1723),  the  celebrated  architect 
of  St.  Paul's  Cathedral  in  London,  were  the  rectification  of  a  cycloidal 
arc  and  the  determination  of  its  centre  of  gravity.  Fermat  found  the 
area  generated  by  an  arc  of  the  cycloid.  Huygens  invented  the  cy- 
cloidal pendulum. 

The  beginning  of  the  seventeenth  century  witnessed  also  a  revival  of 
synthetic  geometry.  One  who  treated  conies  still  by  ancient  methods, 
but  who  succeeded  in  greatly  simplifying  many  prolix  proofs  of  Apollo- 
nius,  was  Claude  Mydorge  (i585-i647J,in  Paris,  a  friend  of  Descartes. 
But  it  remained  for  Girard  Desargues  (1593-1662)  of  Lyons,  and  for 
Pascal,  to  leave  the  beaten  track  and  cut  out  fresh  paths.  They  intro- 
duced the  important  method  of  Perspective.  All  conies  on  a  cone  with 
circular  base  appear  circular  to  an  eye  at  the  apex.  Hence  Desargues 
and  Pascal  conceived  the  treatment  of  the  conic  sections  as  projections 
of  circles.  Two  important  and  beautiful  theorems  were  given  by  Des- 
argues: The  one  is  on  the  "involution  of  the  six  points,"  in  which  a 
transversal  meets  a  conic  and  an  inscribed  quadrangle;  the  other  is 
that,  if  the  vertices  of  two  triangles,  situated  either  in  space  or  in 
a  plane,  lie  on  three  lines  meeting  in  a  point,  then  their  sides  meet  in 
three  points  lying  on  a  line;  and  converselv.  This  last  theorem  has 
been  employed  in  recent  times  by  Brianchon,  C.  Sturm,  Gergonne, 
and  Poncelet.  Poncelet  made  it  the  basis  of  his  beautiful  theory  of 
homological  figures.  We  owe  to  Desargues  the  theory  of  involution 
fe,nd  of  transversals;  also  the  beautiful  conception  that  the  two  ex- 
tremities of  a  straight  line  may  be  considered  as  meeting  at  infinity, 
and  that  parallels  differ  from  other  pairs  of  lines  only  in  having  their 
points  of  intersection  at  infinity.  He  re-invented  the  epicycloid  and 
showed  its  application  to  the  construction  of  gear  teeth,  a  subject 
elaborated  more  fully  later  by  La  Hire.  Pascal  greatly  admired 
Desargues'  results,  saying  (in  his  Essais  pour  les  Coniques),  "I  wish  to 
acknowledge  that  I  owe  the  little  that  I  have  discovered  on  this  sub- 
ject, to  his  writings."  Pascal's  and  Desargues'  writings  contained 
some  of  the  fundamental  ideas  of  modern  synthetic  geometry.  In 
Pascal's  wonderful  work  on  conies,  written  at  the  age  of  sixteen  and 
now  lost,  were  given  the  theorem  on  the  anharmonic  ratio,  first  found 
in  Pappus,  and  also  that  celebrated  proposition  on  the  mystic  hexagon, 
known  as  "Pascal's  theorem,"  viz.  that  the  opposite  sides  of  a  hexa- 
gon inscribed  in  a  conic  intersect  in  three  points  which,  are  collinear. 
This  theorem  formed  the  keystone  to  his  theory.  He  himself  said 
that  from  this  alone  he  deduced  over  400  corollaries,  embracing  the 
conies  of  Apollonius  and  many  other  results.  Less  gifted  than  Des- 
argues and  Pascal  was  Philippe  de  la  Hire  (1640-171S).  At  first 
active  as  a  painter,  he  afterwards  devoted  himself  to  astronomy  and 
mathematics,  and  became  professor  of  the  College  de  France  in  Paris. 
He  wrote  three  works  on  conic  sections,  published  in  1673,  1679  and 


VIETA  TO  DESCARTES  167 

1685.  The  last  of  these,  the  Sediones  Conicae,  was  best  known.  La 
Hire  gave  the  polar  properties  of  circles,  and,  by  projection,  transferred 
his  polar  theory  from  the  circle  to  the  conic  sections.  In  the  construc- 
tion of  maps  De  la  Hire  used  "globular"  projection  in  which  the  eye 
is  not  at  the  pole  of  the  sphere,  as  in  the  Ptolemaic  stereographic  pro- 
jection, but  on  the  radius  produced  through  the  pole  at  a  distance 
r  sin  45°  outside  the  sphere.  Globular  projection  has  the  advantage 
that  everj-^vhere  on  the  map  there  is  approximately  the  same  degree 
of  exaggeration  of  distances.  This  mode  of  projection  was  modified 
by  his  countrjTnan  A.  Parent.  De  la  Hire  wrote  on  roulettes,  on 
graphic  methods,  epicycloids,  conchoids,  and  on  magic  squares.  The 
labors  of  De  la  Hire,  the  genius  of  Desargues  and  Pascal,  uncovered 
several  of  the  rich  treasures  of  modern  synthetic  geometry;  but  owing 
to  the  absorbing  interest  taken  in  the  analytical  geometry  of  Descartes 
and  later  in  the  differential  calculus,  the  subject  was  almost  entirely 
neglected  until  the  nineteenth  century. 

In  the  theory  of  numbers  no  new  results  of  scientific  value  had  been 
reached  for  over  1000  years,  extending  from  the  times  of  Diophantus 
and  the  Hindus  until  the  beginning  of  the  seventeenth  century.  But 
the  illustrious  period  we  are  now  considering  produced  men  who 
rescued  this  science  from  the  realm  of  mysticism  and  superstition, 
in  which  it  had  been  so  long  imprisoned;  "the  properties  of  numbers 
began  again  to  be  studied  scientifically.  Not  being  in  possession  of 
the  Hindu  indeterminate  analysis,  many  beautiful  results  of  the 
Brahmins  had  to  be  re-discovered  by  the  Europeans.  Thus  a  solution 
in  integers  of  linear  indeterminate  equations  was  re-discovered  by  the 
Frenchman  Bachet  de  Meziriac  (1581-1638),  who  was  the  earhest 
noteworthy  European  Diophantist.  In  161 2  he  published  Ptoblemes 
plaisanls  e't  dclectables  qui  se  font  par  les  nombres,  and  in  162 1  a  Greek 
edition  of  Diophantus  with  notes.  An  interest  in  prime  numbers  is 
disclosed  in  the  so-called  "Mersenne's  numbers,"  of  the  form  Mp= 
2P  — I,  with  p  prime.  Marin  INIersenne  asserted  in  the  preface  to  his 
Cogitata  Physico-Mathenialica,  1644,  that  the  only  values  of  p  not 
greater  than  257  which  make  ]\Ip  a  prime  are  i,  2,  3,  5,  7,  13,  17,  19, 
31,  67,  127,  and  257.  Four  mistakes  have  now  been  detected  in 
Alersenne's  classification,  viz.,  Me?  is  composite;  Mei,  Msg  and  M107 
are  prime.  Misi  has  been  found  to  be  composite.  Mersenne  gave  in 
1644  also  the  first  eight  perfect  numbers  6,  28,  496,  8128,  23550336, 
8589869056,  137438691328,  2305843008139952128.  In  Euclid's  Ele- 
ments,  Bk.  9,  Prop.  36,  is  given  the  formula  for  perfect  numbers 
2P^-i(2P—  i),  where  2P~^— i  is  prime.  The  above  eight  perfect  numbers 
are  reproduced  by  taking  p  =  2,  t,,  5,  7,  13,  17,  19,  31.  A  ninth  perfect 
number  was  found  in  1885  by  P.  Seelhoff,  for  which  p=6i,  a  tenth 
in  191 2  by  R.  E.  Powers,  for  which  p=8g.  The  father  of  the  modern 
theory  of  numbers  is  Fermat.  He  was  so  uncommunicative  in  dis- 
position, that  he  generally  concealed  his  methods  and  made  known 


i68  A  HISTORY  OF  MATHEMATICS 

his  results  only.  In  some  cases  later  analysts  have  been  greatly 
puzzled  in  the  attempt  of  supplying  the  proofs.  Fermat  owned  a  copy 
of  Bachet's  Diophantus,  in  which  he  entered  numerous  marginal  notes. 
In  1670  these  notes  were  incorporated  in  a  new  edition  of  Diophantus, 
brought  out  by  his  son.  Other  theorems  on  numbers,  due  to  Fermat, 
were  published  in  his  Opera  varia  (edited  by  his  son)  and  in  Wallis's 
Commerciuni  epistolicum  of  1658.  Of  the  following  theorems,  the 
first  seven  are  found  in  the  marginal  notes:  ^ — 

(i)  x"'+y*'  =  z"  is  impossible  for  integral  values  of  x,  y,  and  z,  when 
n>2. 

This  famous  theorem  was  appended  by  Fermat  to  the  problem  of 
Diophantus  II,  8 :  "  To  divide  a  given  square  number  into  two  squares." 
Fermat's  marginal  note  is  as  follows:  "On  the  other  hand  it  is  im- 
possible to  separate  a  cube  into  two  cubes,  or  a  biquadrate  into  two 
biquadrates,  or  generally  any  power  except  a  square  into  two  powers 
with  the  same  exponent.  I  have  discovered  a  truly  marvelous  proof 
of  this,  which  however  the  margin  is  not  large  enough  to  contain." 
That  Fermat  actually  possessed  a  proof  is  doubtful.  No  general 
proof  has  yet  been  published.  Euler  proved  the  theorem  for  w=3 
and  w=4;  Dirichlet  for  n  =  s  and  w=i4,  G.  Lame  for  n=7  and  Kum- 
mer  for  many  other  values.  Repeatedly  was  the  theorem  made  the 
prize  question  of  learned  societies,  by  the  Academy  of  Sciences  in 
Paris  in  1823  and  1850,  by  the  Academy  of  Brussels  in  1883.  The 
recent  history  of  the  theorem  follows  later. 

(2)  A  prime  of  the  form  4W+1  is  only  once  the  hypothenuse  of  a 
right  triangle;  its  square  is  twice;  its  cube  is  three  times,  etc.  Ex- 
ample: 52=32+42;  252=152+202=72+242;  1252=752+1002=352+ 
1202=442+1172. 

(3)  A  prime  of  the  form  4»+i  can  be  expressed  once,  and  only 
once,  as  the  sum  of  two  squares.    Proved  by  Euler. 

(4)  A  number  composed  of  two  cubes  can  be  resolved  into  two 
other  cubes  in  an  infinite  multiplicity  of  ways. 

(5)  Every  number  is  either  a  triangular  number  or  the  sum  of  two 
or  three  triangular  numbers;  either  a  square  or  the  sum  of  two,  three, 
or  four  squares;  either  a  pentagonal  number  or  the  sum  of  two,  three, 
four,  or  five  pentagonal  numbers;  similarly  for  polygonal  numbers 
in  general.  The  proof  of  this  and  other  theorems  is  promised  by 
Fermat  in  a  future  work  which  never  appeared.  This  theorem  is 
also  given,  with  others,  in  a  letter  of  1637  (?)  addressed  to  Pater 
Mersenne. 

(6)  As  many  numbers  as  you  please  may  be  found,  such  that  the 
square  of  each  remains  a  square  on  the  addition  to  or  subtraction  from 
jt  of  the  sum  of  all  the  numbers. 

1  For  a  fuller  historical  account  oi  Fermat's  Diophantine  theorems  and  prob- 
lems, see  T.  L.  Heath,  Diophantus  of  Alexandria,  2.  Ed.,  1910,  pp.  267-328.  See 
also  Annals  oj  Mathematics,  2.  S.,  Vol.  18,  19171  PP-  161-187. 


VIE  FA  TO  DESCARTES  i6Q 

(7)  .r''+V*^  =  -"  is  impossible. 

(8)  In  a  letter  of  1640  he  gives  the  celebrated  ;heorem  generally 
known  as  "Fermat's  theorem,"  which  we  state  in  Gauss's  notation: 
If  p  is  prime,  and  a  is  prime  to  p,  then  a^~^~i  (mod  p).  It  was  proved 
by  Leibniz  and  by  Euler. 

(9)  Fermat  died  with  the  belief  that  he  had  found  a  long-sought-for 
law  of  prime  numbers  in  the  formula  22"+i=a  prime,  but  he  admitted 
that  he  was  unable  to  prove  it  rigorously.  The  law  is  not  true,  as  was 
pointed  out  by  Euler  in  the  example  22'''+i=4, 294,967, 297  =  6, 700,417 
times  641.  The  American  lightning  calculator  Zerali  Colbuni,  when 
a  boy,  readily  found  the  factors,  but  was  unable  to  explain  the  method 
by  which  he  made  his  marvellous  mental  computation. 

(10)  An  odd  prime  number  can  be  expressed  as  the  difference  of 
two  squares  in  one,  and  only  one,  way.  This  theorem,  given  in  the 
Relation,  was  used  by  Fermat  for  the  decomposition  of  large  numbers 
into  prime  factors. 

(11)  If  the  integers  a,  b,  c  represent  the  sides  of  a  right  triangle, 
then  its  area  cannot  be  a  square  number.  This  was  proved  by  La- 
grange. 

(12)  Fermat's  solution  of  a.r-+i=y",  where  a  is  integral  but  not 
a  square,  has  come  down  in  only  the  broadest  outline,  as  given  in  the 
Relation.  He  proposed  the  problem  to  the  Frenchman,  Bernhard 
Frenide  dc  Bessy,  and  in  1657  to  all  living  mathematicians.  In  Eng- 
land, Wallis  and  Lord  Brouncker  conjointly  found  a  laborious  solution, 
which  was  published  in  1658,  and  also  in  166S  in  Thomas  Erancker's 
translation  of  Rahn's  Algebra,  "altered  and  augmented"  by  John 
Pell  (1610-16S5).  The  first  solution  was  given  by  the  Hindus. 
Though  Pell  had  no  other  connection  with  the  problem,  it  went  by 
the  name  of  "  Pell's  problem."  Pell  held  at  one  time  the  mathematical 
chair  at  Amsterdam.  In  a  controversy  with  Longomontanus  who 
claimed  to  have  effected  the  quadrature  of  the  circle.  Pell  first  used 
the  now  famihar  trigonometric  formula  tan2.4  =  2tan/l/(i  — tan-^). 

We  are  not  sure  that  Fermat  subjected  all  his  theorems  to  rigorous 
proof.  His  methods  of  proof  were  entirely  lost  until  1879,  when  a 
document  was  found  buried  among  the  manuscripts  of  Huygens  in 
the  library  of  Leyden,  entitled  Relation  des  decouvcrtes  en  la  science  des 
nombres.  It  appears  from  it  that  he  used  an  inductive  method,  called 
by  him  la  descentc  infinie  on  indefinie.  He  says  that  this  was  particu- 
larly applicable  in  proving  the  impossibility  of  certain  relations,  as, 
for  instance,  Theorem  11,  given  above,  but  that  he  succeeded  in  using 
the  method  also  in  proving  affirmative  statements.  Thus  he  proved 
Theorem  3  by  showing  that  if  we  suppose  there  be  a  prime  pi+i 
which  does  not  possess  this  property,  then  there  will  be  a  smaller 
prime  of  the  form  4n+i  not  possessing  it;  and  a  third  one  smaller 
than  the  second,  not  possessing  it;  and  so  on.  Thus  descending  in- 
definitely, he  arrives  at  the  number  5,  which  is  the  smallest  jirime 


170  A  HISTORY  OF  MATHEMATICS 

factor  of  the  form  ^n+i.  From  the  above  supposition  it  would  follow 
that  5  is  not  the  sum  of  two  squares— a  conclusion  contrary  to  fact. 
Hence  the  supposition  is  false,  and  the  theorem  is  established.  Fermat 
appUed  this  method  of  descent  with  success  in  a  large  number  of 
theorems.  By  this  method  L.  Euler,  A.  M.  Legendre,  P.  G.  L.  Dirich- 
let,  proved  several  of  his  enunciations  and  many  other  numerical 
propositions. 

Fermat  was  interested  in  magic  squares.  These  squares,  to  which 
the  Chinese  and  Arabs  were  so  partial,  reached  the  Occident  not  later 
than  the  fifteenth  century.  A  magic  square  of  25  cells  was  found  by 
M.  Curtze  in  a  German  manuscript  of  that  time.  The  artist,  Albrecht 
Dijrer,  exhibits  one  of  16  cells  in  1514  in  his  painting  called  "Melan- 
chohe."  The  above-named  Bernhard  Frenide  de  Bessy  (about  1602- 
1675)  brought  out  the  fact  that  the  number  of  magic  squares  increased 
enormously  with  the  order  by  writing  down  8S0  magic  squares  of 
the  order  four.  Fermat  gave  a  general  rule  for  finding  the  number  of 
magic  squares  of  the  order  n,  such  that,  for  w=8,  this  number  was 
1,004,144,995,344;  but  he  seems  to  have  recognized  the  falsity  of  his 
rule.  Bachet  de  Meziriac,  in  his  Problemes  plaisants  et  delectahles, 
Lyon,  1612,  gave  a  rule  "des  terrasses"  for  writing  down  magic 
squares  of  odd  order.  Frenicle  de  Bessy  gave  a  process  for  those  of 
even  order.  In  the  seventeenth  century  magic  squares  were  studied  ^ 
by  Antoine  Arnauld,  Jean  Prestet,  J.  Ozanam;  in  the  eighteenth  cen- 
tury by  Poignard,  De  la  Hire,  J.  Sauveur,  L.  L.  Pajot,  J.  J.  Rallier 
des  Ourmes,  L.  Euler  and  Benjamin  Franklin.  In  a  letter  B.  FrankUn 
said  of  his  magic  square  of  16''  cells,  "I  make  no  question,  but  you 
will  readily  allow  the  square  of  16  to  be  the  most  magically  magical 
of  any  magic  square  ever  made  by  any  magician." 

A  correspondence  between  B.  Pascal  and  P.  Fermat  relating  to  a 
certain  game  of  chance  was  the  germ  of  the  theory  of  probabilities, 
of  which  some  anticipations  are  found  in  Cardan,  Tartaglia,  J.  Kepler 
and  Galileo.  Chevalier  de  Mere  proposed  to  B.  Pascal  the  funda- 
mental "Problem  of  Points,"^  to  determine  the  probability  which 
each  player  has,  at  any  given  stage  of  the  game,  of  winning  the  game. 
Pascal  and  Fermat  supposed  that  the  players  have  equal  chances  of 
winning  a  single  point. 

The  former  communicated  this  problem  to  Fermat,  who  studied 
it  with  lively  interest  and  solved  it  by  the  theory  of  combinations,  a 
theory  which  was  diligently  studied  both  by  him  and  Pascal.  The 
calculus  of  probabilities  engaged  the  attention  also  of  C.  Huygens. 
The  most  important  theorem  reached  by  him  was  that,  if  A  has  p 
chances  of  winning  a  sum  a,  and  q  chances  of  winning  a  sum  b,  then 

1  Encvclopedie  des  sciences  math's,  T.  T,  Vol.  3,  1906,  p.  66. 

"^Oeu'vrcs  completes  de  Blaise  Pascal,  T.  I,  Paris,  1S66,  pp.  220-237.  See  also  I. 
Todhunter,  History  of  the  Mathematical  Theory  of  Prohabiiily,  Cambridge  and 
London,  1865,  Chapter  II. 


VIETA  TO  DESCARTES  171 

he  may  expec^  to  win  the  sum  -^ — -.    Huygens  gave  his  results  in 

a  treatise  on  probability  (1657),  which  was  the  best  account  of  the 
subject  until  the  appearance  of  Jakob  Bernoulli's  Ars  conjedamii 
which  contained  a  reprint  of  Huygens'  treatise.  An  absurd  abuse  of 
mathematics  in  connection  with  tlie  probability  of  testimony  was 
made  by  John  Craig  who  in  1699  concluded  that  faith  in  the  Gospel 
so  far  as  it  depended  on  oral  tradition  expired  about  the  year  800, 
and  so  far  as  it  depended  on  written  tradition  it  would  expire  in  the 
year  3150. 

Connected  with  the  theory  of  probabiHty  were  the  investigations 
on  mortality  and  insurance.  The  use  of  tables  of  mortahty  does  not 
seem  to  have  been  altogether  unknown  to  the  ancients,  but  the  first 
name  usually  mentioned  in  this  connection  is  Captain  John  Graunt 
who  pubhshed  at  London  in  1662  his  Natural  and  Political  Observa- 
tions .  .  .  made  upon  the  bills  of  mortality,  basing  his  deductions  upon 
records  of  deaths  which  began  to  be  kept  in  London  in  1592  and  were 
first  intended  to  make  known  the  progress  of  the  plague.  Graunt  was 
careful  to  pubhsh  the  actual  figures  on  which  he  based  his  conclusions, 
comparing  himself,  when  so  doing,  to  a  "silly  schoolboy,  coming  to 
say  his  lessons  to  the  world  (that  peevish  and  tetchie  master),  who 
brings  a  bundle  of  rods,  wherewith  to  be  whipped  for  every  mistake 
he  has  committed."  ^  Nothing  of  marked  importance  was  done  after 
Graunt  until  1693  when  Edmund  Halley  ^  published  in  the  Philo- 
sophical Transactions  (London)  his  celebrated  memoir  on  the  Degrees 
of  Mortality  of  Mankind  .  .  .  with  an  Atte?npi  to  ascertain  the  Price  of 
Annuities  upon  Lives.  To  find  the  value  of  an  annuity,  multiply  the 
chance  that  the  individual  concerned  will  be  alive  after  n  years  by 
the  present  value  of  the  annual  payment  due  at  the  end  of  »  years; 
then  sum  the  results  thus  obtained  for  all  values  of  n  from  i  to  the 
extreme  possible  age  for  the  life  of  that  individual.  Halley  considers 
also  annuities  on  joint  lives. 

Among  the  ancients,  Archimedes  was  the  only  one  who  attained 
clear  and  correct  notions  on  theoretical  statics.  He  had  acquired 
firm  possession  of  the  idea  of  pressure,  which  lies  at  the  root  of  me- 
chanical science.  But  his  ideas  slept  nearly  twenty  centuries,  until 
the  time  of  S.  Stevin  and  Galileo  Galilei  (1564-1642).  Stevin  deter- 
mined accurately  the  force  necessary  to  sustain  a  body  on  a  plane 
inclined  at  any  angle  to  the  horizon.  He  was  in  possession  of  a  com- 
plete doctrine  of  equilibrium.  While  Stevin  investigated  statics, 
Galileo  pursued  principally  dynamics.  Galileo  was  the  first  to  abandon 
the  idea  usually  attributed  to  Aristotle  that  bodies  descend  more 
quickly  in  proportion  as  they  are  heavier;  he  established  the  first  law 
of  motion;  determined  the  laws  of  falling  bodies;  and,  having  obtained 

'  I.  Todhunter,  History  of  the  Theory  of  Frobabilily,  pp.  38,  42. 


172  A  HISTORY  OF  MATHEMATICS 

a  clear  notion  of  acceleration  and  of  the  independence  of  different 
.notions,  was  able  to  prove  that  projectiles  move  in  parabolic  curves. 
Up  to  his  time  it  was  believed  that  a  cannon-ball  moved  forward  at 
first  in  a  straight  hne  and  then  suddenly  fell  vertically  to  the  ground. 
Galileo  had  an  understanding  of  centrifugal  forces,  and  gave  a  correct 
definition  of  momentum.  Though  he  formulated  the  fundamental 
principle  of  statics,  known  as  the  parallelogram  of  forces,  yet  he  did 
not  fully  recognize  its  scope.  The  principle  of  virtual  velocities  was 
partly  conceived  by  Guido  Ubaldo  (died  1607),  and  afterwards  more 
fully  by  Galileo. 

Galileo  is  the  founder  of  the  science  of  dynamics.  Among  his  con- 
temporaries it  was  chiefly  the  novelties  he  detected  in  the  sky  that 
made  him  celebrated,  but  J.  Lagrange  claims  that  his  astronomical 
discoveries  required  only  a  telescope  and  perseverance,  while  it  took 
an  extraordinary  genius  to  discover  laws  from  phenomena,  which  we 
see  constantly  and  of  which  the  true  explanation  escaped  all  earlier 
philosophers.  Galileo's  dialogues  on  mechanics,  the  Discorsi  e  demos- 
irazioni  matematiclie,  1638,  touch  also  the  subject  of  infinite  aggregates. 
The  author  displays  a  keenness  of  vision  and  an  originality  which 
was  not  equalled  before  the  time  of  Dedekind  and  Georg  Cantor. 
Salviati,  who  in  general  represents  Galileo's  own  ideas  in  these  dia- 
logues, says,^  "infinity  and  indivisibility  are  in  their  very  nature  in- 
comprehensible to  us."  Simplicio,  who  is  the  spokesman  of  Aris- 
totelian scholastic  philosophy,  remarks  that  "the  infinity  of  points 
in  the  long  line  is  greater  than  the  infinity  of  points  in  the  short  line." 
Then  come  the  remarkable  words  of  Salviati:  "This  is  one  of  the 
difficulties  which  arise  when  we  attempt,  with  our  finite  minds,  to 
discuss  the  infinite,  assigning  to  it  those  properties  which  we  give  to 
the  finite  and  unlimited;  but  this  I  think  is  wrong,  for  we  cannot 
speak  of  infinite  quantities  as  being  the  one  greater  or  less  than  or 
equal  to  another.  .  .  .  We  can  only  infer  that  the  totality  of  all 
numbers  is  infinite,  and  that  the  number  of  squares  is  infinite,  and 
that  the  number  of  the  roots  is  infinite;  neither  is  the  number  of  squares 
less  than  the  totality  of  all  numbers,  nor  the  latter  greater  than  the 
former;  and  finally  the  attributes  'equal,'  'greater,'  and  'less'  are 
not  applicable  to  infinite,  but  only  to  finite  quantities.  .  .  .  One 
line  does  not  contain  more  or  less  or  just  as  many  points  as  another, 
but  .  .  .  each  line  contains  an  infinite  number."  From  the  time  of 
Galileo  and  Descartes  to  Sir  William  Hamilton,  there  was  held  the 
doctrine  of  the  finitude  of  the  human  mind  and  its  consequent  in- 
ability to  conceive  the  infinite.  A.  De  Morgan  ridiculed  this,  saying, 
the  argument  amounts  to  this,  "who  drives  fat  oxen  should  himself 
be  fat." 

Infinite  series,  which  sprang  into  prominence  at  the  time  of  the 

1  See  Galileo's  Dialogues  concerning  two  new  Sciences,  translated  by  Henry  Crew 
and  Alfonso  de  Salvio,  New  York,  1914,  "First  Day,"  pp.  30-32. 


DESCARTES  TO  NEWTON  173 

invention  of  the  differential  and  integral  calculus,  were  used  by  a  few 
writers  before  that  time.  Pielro  Mengoli  (1626-1686)  of  Bologna  ' 
treats  them  in  a  book,  Noi<z  quadraluiw  anlhmclicce,  of  1650.  He 
proves  the  divergence  of  the  harmonic  series  by  dividing  its  terms 
into  an  infinite  number  of  groups,  such  that  the  sum  of  the  terms  in 
each  group  is  greater  than  i.  The  first  proof  of  this  was  formerly 
attributed  to  Jakob  Bernoulli,  1689.  Mengoli  showed  the  conver- 
gence of  the  reciprocals  of  the  triangular  numbers,  a  result  formerly 
supposed  to  have  been  first  reached  by  C.  Huygcns,  G.  W.  Leibniz, 
or  Jakob  Bernoulli.  MengoU  reached  creditable  results  on  the  sum- 
mation of  infinite  series. 

Dcscarles  to  Newton 

Among  the  earliest  thinkers  of  the  seventeenth  and  eighteenth 
centuries,  who  employed  their  mental  powers  toward  the  destruction 
of  old  ideas  and  the  up-building  of  new  ones,  ranks  Rene  Descartes 
(1596-1650).  Though  he  professed  orthodoxy  in  faith  all  his  life, 
yet  in  science  he  was  a  profound  sceptic.  He  found  that  the  world's 
brightest  thinkers  had  been  long  exercised  in  metaphysics,  yet  they 
had  discovered  nothing  certain;  nay,  had  even  flatly  contradicted  each 
other.  This  led  him  to  the  gigantic  resolution  of  taking  nothing 
whatever  on  authority,  but  of  subjecting  everything  to  scrutinous 
examination,  according  to  new  methods  of  inquiry.  The  certainty  of 
the  conclusions  in  geometry  and  arithmetic  brought  out  in  his  mind 
the  contrast  between  the  true  and  false  ways  of  seeking  the  truth. 
He  thereupon  attempted  to  apply  mathematical  reasoning  to  all 
sciences.  ''Comparing  the  mysteries  of  nature  with  the  laws  of 
mathematics,  he  dared  to  hope  that  the  secrets  of  both  could  be  un- 
locked with  the  same  key,"  Thus  he  built  up  a  system  of  philosophy 
called  Cartesianism. 

Great  as  was  Descartes'  celebrity  as  a  metaphysician,  it  may  be 
fairly  questioned  whether  his  claim  to  be  remembered  by  posterity 
as  a  mathematician  is  not  greater.  His  philosophy  has  long  since 
been  superseded  by  other  systems,  but  the  analytical  geometry  of 
Descartes  will  remain  a  valuable  possession  forever.  At  the  age  of 
twenty-one,  Descartes  enUsted  in  the  army  of  Prince  Maurice  of 
Orange.  His  years  of  soldiering  were  years  of  leisure,  in  which  he  had 
time  to  pursue  his  studies.  At  that  time  mathematics  was  his  favorite 
science.  But  in  1625  he  ceased  to  devote  himself  to  pure  mathe- 
matics.    Sir  William  Hamilton  ^  is  in   error  when  he  states  that 

'  See  G.  Encstrom  in  BibJiothcca  mathrmalica,  3.  S.,  Vol.  12,  igii-12,  pp.  135-148. 

"^  Sir  William  Hamilton,  the  metai)hysician,  made  a  famous  attack  upon  the 
study  of  mathematics  as  a  traininj^  of  the  mind,  which  apj^cared  in  the  Edinburgh 
Kcvicw  of  1836.  It  was  shown  by  A.  T.  Bledsoe  in  the  Soullu-ni  Review  for  July, 
1877,  that  Hamilton  misrepresented  the  sentiments  held  by  Descartes  and  other 
scientists.    See  also  J.  S.  Mill's  Examinalion  oj  Sir  William  Hamilton's  Philosophy; 


174  A  HISTORY  OF  MATHEMATICS 

Descartes  considered  mathematical  studies  absolutely  pernicious  as  a 
means  of  internal  culture.  In  a  letter  to  Mersenne,  Descartes  says: 
"M.  Desargues  puts  me  under  obligations  on  account  of  the  pains 
that  it  has  pleased  him  to  have  in  me,  in  that  he  shows  that  he  is 
sorry  that  I  do  not  wish  to  study  more  in  geometry,  but  I  have  re- 
solved to  quit  only  abstract  geometry,  that  is  to  say,  the  consideration 
of  questions  which  serve  only  to  exercise  the  mind,  and  this,  in  order  to 
study  another  kind  of  geometry,  which  has  for  its  object  the  explana- 
tion of  the  phenomena  of  nature.  .  .  .  You  know  that  all  my  physics 
is  nothing  else  than  geometry."  The  years  between  1629  and  1649 
were  passed  by  him  in  Holland  in  the  study,  principally,  of  physics 
and  metaphysics.  His  residence  in  Holland  was  during  the  most 
brilliant  days  of  the  Dutch  state.  In  1637  he  pubHshed  his  Discours 
de  la  Methode,  containing  among  others  an  essay  of  106  pages  on 
geometry.  His  Geometric  is  not  easy  reading.  An  edition  appeared 
subsequently  with  notes  by  his  friend  De  Beaune,  which  were  intended 
to  remove  the  difficulties.  The  Geometrie  of  Descartes  is  of  epoch- 
making  importance;  nevertheless  we  cannot  accept  Michel  Chasles' 
statement  that  this  work  is  proles  sine  malre  creata — a  child  brought 
into  being  without  a  mother.  In  part,  Descartes'  ideas  are  found  in 
Apollonius;  the  application  of  algebra  to  geometry  is  found  in  Vieta, 
Ghetaldi,  Oughtred,  and  even  among  the  Arabs.  Fermat,  Descartes' 
contemporary,  advanced  ideas  on  analytical  geometry  akin  to  his 
own  in  a  treatise  entitled  Ad  locos  pianos  et  solidos  isagoge,  which, 
however,  was  not  published  until  1679  in  Fermat's  Varia  opera.  In 
Descartes'  Geometrie  there  is  no  systematic  development  of  the 
method  of  analytics.  The  method  must  be  constructed  from  isolated 
statements  occurring  in  different  parts  of  the  treatise.  In  the  32 
geometric  drawings  illustrating  the  text  the  axes  of  coordinates  are 
in  no  case  expUcitly  set  forth.  The  treatise  consists  of  three  "books." 
The  first  deals  with  "problems  which  can  be  constructed  by  the  aid 
of  the  circle  and  straight  fine  only."  The  second  book  is  "on  the 
nature  of  curved  lines."  The  third  book  treats  of  the  "construction 
of  problems  solid  and  more  than  soKd."  In  the  first  book  it  is  made 
clear,  that  if  a  problem  has  a  finite  number  of  solutions,  the  final 
equation  obtained  will  have  only  one  unknown,  that  if  the  final 
equation  has  two  or  more  unknowns,  the  problem  "is  not  wholly 
determined."  ^  If  the  final  equation  has  two  unknowns  "then  since 
there  is  always  an  infinity  of  different  points  which  satisfy  the  de- 
mand, it  is  therefore  required  to  recognize  and  trace  the  line  on  which 
all  of  them  must  be  located"  (p.  9).    To  accompHsh  this  Descartes 

C.  J.  Keyser,  Mathematics,  1907,  pp.  20-44;  F.  Cajori  in  Popular  Science  Monthly^ 
1912,  pp.  360-372. 

^  Descartes'  Geometrie,  ed.  1886,  p.  4.  We  are  here  guided  by  G.  Enestrom  iii 
Bibliotheca  mathematica,  3.  S.,  Vol.  11,  pp.  240-243;  Vol.  12,  pp.  273,  274;  Vol.  14, 
p.  357,  and  by  H.  Wieleitner  in  Vol.  14,  pp.  241-243,  329,  330. 


DKSCARTES  TO  NFAVTON  17  c; 

sclecLs  a  straight  lino  which  he  sometimes  calls  a  "diameter"  (p.  31) 
and  associates  each  of  its  points  with  a  point  sought  in  such  a  way  that 
the  latter  can  be  constructed  when  the  former  point  is  assumed  as 
known.  Thus,  on  p.  18  he  says,  "Je  choises  une  ligne  droite  comma 
AB,  pour  rapporter  a  ses  divers  points  tous  ceux  de  cette  ligne  courbe 
A'C."  Here  Descartes  follows  Apollonius  who  related  the  points  of 
a  conic  to  the  points  of  a  diameter,  by  distances  (ordinatcs)  which 
make  a  constant  angle  with  the  diameter  and  are  determined  in  length 
by  the  position  of  the  point  on  the  diameter.  This  constant  angle  is 
with  Descartes  usually  a  right  angle.  The  new  feature  introduced  by 
Descartes  was  the  use  of  an  equation  with  more  than  one  unknown,  so 
that  (in  case  of  two  unknowns)  for  any  value  of  one  unknown  (ab- 
scissa), the  length  of  the  other  (ordinate)  could  be  computed.  He 
uses  the  letters  x  and  y  for  the  abscissa  and  ordinate.  He  makes  it 
l)Iain  that  the  x  and  y  may  be  represented  by  other  distances  than  the 
ones  selected  by  him  (p.  10),  that,  for  instance,  the  angle  formed  by 
.r  and  y  need  not  be  a  right  angle.  It  is  noteworthy  that  Descartes 
and  Fermat,  and  their  successors  down  to  the  middle  of  the  eighteenth 
century,  used  oblique  coordinates  more  frequently  than  did  later 
analysts.  It  is  also  noteworthy  that  Descartes  does  not  formally 
introduce  a  second  axis,  our  y-axis.  Such  formal  introduction  is  found 
in  G.  Cramer's  Introduction  d  Vanalysc  des  ligncs  courbcs  algebriques, 
1750;  earlier  publications  by  de  Gua,  L.  Euler,  W.  Murdoch  and  others 
contain  only  occasional  references  to  a  y-axis.  The  words  "abscissa," 
"ordinate"  were  not  used  by  Descartes.  In  the  strictly  technical 
sense  of  analytics  as  one  of  the  coordinates  of  a  point,  the  word 
"ordinate"  was  used  by  Leibniz  in  1694,  but  in  a  less  restricted  sen.se 
such  expressions  as  "ordinatim  applicatae"  occur  much  earlier  in 
F.  Commandinus  and  others.  The  technical  use  of  "abscissa"  is 
observed  in  the  eighteenth  century  by  C.  Wolf  and  others.  In  the 
more  general  sense  of  a  "distance"  it  was  used  earlier  by  B.  Cavalieri 
in  his  Indivisibles,  by  Stefano  dcgli  Angcli  (1623-1697),  a  professor 
of  mathematics  in  Rome,  and  by  others.  Leibniz  introduced  the  word 
"  coordinatae  "  in  1692.  To  guard  against  certain  current  historical 
errors  we  quote  the  following  from  P.  Tannery:  "One  frequently 
attributes  wTongly  to  Descartes  the  introduction  of  the  convention 
of  reckoning  coordinates  positively  and  negatively,  in  the  sense  in 
which  we  start  them  from  the  origin.  The  truth  is  that  in  this  respect 
the  Geometrie  of  1637  contains  only  certain  remarks  touching  the 
interpretation  of  real  or  false  (positive  or  negative)  roots  of  equations. 
" .  .  .If  then  we  examine  with  care  the  rules  given  by  Descartes  in 
his  Geometrie,  as  well  as  his  application  of  them,  we  notice  that  he 
adopts  as  a  principle  that  an  equation  of  a  geometric  locus  is  not 
valid  except  for  the  angle  of  the  coordinates  (quadrant)  in  which  it 
was  established,  and  all  his  contemporaries  do  likewise.  The  extension 
of  an  equation  to  other  angles  (quadrants)  was  freely  made  in  particu- 


176  A  HISTORY  OF  MATHEMATICS 

iar  cases  for  the  interpretation  of  the  negative  roots  oi  equations; 
but  while  it  served  particular  conventions  (for  example  for  reckoning 
distances  as  positive  and  negative),  it  was  in  reality  quite  long  in 
completely  establishing  itself,  and  one  cannot  attribute  the  honor  for 
it  to  any  particular  geometer." 

Descartes'  geometry  was  called  "analytical  geometry,"  partly 
because,  unlike  the  synthetic  geometry  of  (he  ancients,  it  is  actually 
analytical,  in  the  sense  that  the  word  is  used  in  logic;  and  partly  be- 
cause the  practice  had  then  already  arisen,  of  designating  by  the  term 
analysis  the  calculus  with  general  quantities 

The  first  important  example  solved  by  Descartes  in  his  geometry 
is  the  "problem  of  Pappus";  viz.  "Given  several  straight  Hues  in  a 
plane,  to  find  the  locus  of  a  point  such  that  the  perpendiculars,  or  more 
generally,  straight  lines  at  given  angles,  drawn  from  the  point  to  the 
given  lines,  shall  satisfy  the  condition  that  the  product  of  certain  of 
them  shall  be  in  a  given  ratio  to  the  product  of  the  rest."  Of  this 
celebrated  problem,  the  Greeks  solved  only  the  special  case  when  the 
number  of  given  lines  is  four,  in  which  case  the  locus  of  the  point 
turns  out  to  be  a  conic  section.  By  Descartes  it  was  solved  com- 
pletely, and  it  afforded  an  excellent  example  of  the  use  which  canbe 
made  of  his  analytical  method  in  the  study  of  loci.  Another  solution 
was  given  later  by  Newton  in  the  Principia.  Descartes  illustrates 
his  analytical  method  also  by  the  ovals,  now  named  after  him,  "cer- 
taines  ovales  que  vous  verrez  etre  tres-utiles  pour  la  theorie  de  la 
catoptrique."  These  curves  were  studied  by  Descartes,  probably,  as 
early  as  1629;  they  were  intended  by  him  to  serve  in  the  construction 
of  converging  lenses,  but  yielded  no  results  of  practical  value.  In 
the  nineteenth  century  they  received  much  attention.^ 

The  power  of  Descartes'  analytical  method  in  geometry  has  been 
vividly  set  forth  recently  by  L.  Boltzmann  in  the  remark  that  the 
formula  appears  at  times  cleverer  than  the  man  who  invented  it.  Of 
all  the  problems  which  he  solved  by  his  geometry,  none  gave  him  as 
great  pleasure  as  his  mode  of  constructing  tangents.  It  was  published 
earlier  than  the  methods  of  Fermat  and  Roberval  which  were  noticed 
on  a  preceding  page. 

Descartes'  method  consisted  in  first  finding  the  normal.  Through 
a  given  point  x,  y  of  the  curve  he  drew  a  circle  which  had  its  centre 
at  the  intersection  of  the  normal  and  the  x-axis.  Then  he  imposed 
the  condition  that  the  circle  cut  the  curve  in  two  coincident  points 
X,  y.  In  1638  Descartes  indicated  in  a  letter  that,  in  place  of  the 
circle,  a  straight  line  may  be  used.  This  idea  is  elaborated  by  Flori- 
mond  de  Beaune  in  his  notes  to  the  1649  edition  of  Descartes'  Geometrie. 
In  finding  the  point  of  intersection  of  the  normal  and  .r-axis,  Descartes 
used  the  method  of  Indeterminate  Coefficients,  of  which  he  bears  the 
honor  of  invention.  Indeterminate  coefficients  were  employed  by 
^  See  G.  Loriu,  Ehenc  Curvcn  (F.  Schutte),  I,  igio,  p.  174. 


DESCARTES  TO  NEWTON  177 

him  also  in  solving  biquadratic  equations.  Descartes'  method  of 
tangents  is  profound,  but  operose,  and  inferior  to  Fermat's  method. 
In  the  third  book  of  his  Geomelrie  he  points  out  that  if  a  cubic  equation 
(with  rational  coefficients)  has  a  rational  root,  then  it  can  be  factored 
and  the  cubic  can  be  solved  geometrically  by  the  use  of  ruler  and 
compasses  only.  He  derives  the  cubic  z^=2)Z  —  q  as  the  equation  upon 
which  the  trisection  of  an  angle  depends.  He  effects  a  trisection  by  the 
aid  of  a  parabola  and  circle,  but  does  not  consider  the  reducibility  of 
the  equation.  Hence  he  left  the  question  of  the  "  insolvability "  of 
the  problem  untouched.  Not  till  the  nineteenth  century  were  con- 
clusive proofs  advanced  of  the  impossibility  of  trisecting  any  angle 
and  of  duplicating  a  cube,  culminating  at  last  in  the  clear  and  simple 
proofs  given  by  F.  Klein  in  1895  in  his  Ansgewahlle  Fragen  der  Elemen- 
targeomctric,  translated  into  English  in  1897  by  W.  W.  Beman  and 
D.  E.  Smith.  Descartes  proved  that  every  geometric  problem  giving 
rise  to  a  cubic  equation  can  be  reduced  either  to  the  duplication  of  a 
cube  or  to  the  trisection  of  an  angle.  This  fact  had  been  previously 
recognized  by  \'ieta. 

The  essays  of  Descartes  on  dioptrics  and  geometry  were  sharply 
criticised  by  Fermat,  who  wrote  objections  to  the  former,  and  sent 
his  own  treatise  on  "maxima  and  minima"  to  show  that  there  were 
omissions  in  the  geometry.  Descartes  thereupon  made  an  attack  on 
Fermat's  method  of  tangents.  Descartes  was  in  the  wrong  in  this 
attack,  yet  he  continued  the  controversy  with  obstinacy.  In  a  letter 
of  163S,  addressed  to  Mersenne  and  to  be  transmitted  to  Fermat, 
Descartes  gives  .v^+y^  =  a.vv,  now  known  as  the  "folium  of  Descartes." 
as  representing  a  curve  to  which  Fermat's  method  of  tangents  would 
not  apply.  ^  The  curve  is  accompanied  by  a  figure  which  shows  that 
Descartes  did  not  then  know  the  shape  of  the  curve.  At  that  time 
the  fundamental  agreement  about  algebraic  signs  of  coordinates  had 
not  yet  been  hit  upon ;  only  finite  values  of  variables  were  used.  Hence 
the  infinite  branches  of  the  curve  remained  unnoticed;  some  investi- 
gators thought  there  were  four  leaves  instead  of  only  one.  C.  Huygens 
in  1692  gave  the  correct  shape  and  the  asymptote  of  the  curve. 

Parabolas  of  higher  order,  y"  =  p"-~^x,  are  mentioned  by  Descartes 
in  a  letter  of  July  13,  163S,  in  which  the  centre  of  mass  and  the  volume 
obtained  by  revolution  are  considered.  Cognate  considerations  are 
due  to  G.  P.  Roberval,  P.  Fermat  and  B.  Cavalieri.  Apparently,  the 
shapes  of  these  curves  were  not  studied,  and  it  remained  for  C.  Mac- 
laurin  (174S)  and  G.  F.  A.  I'Hospital  (1770)  to  remark  that  they 
have  wholly  different  shapes,  according  to  whether  11  is  a  positive  or 
a  negative  integer. 

Descartes  had  a  controversy  with  G.  P.  Roberval  on  the  cycloid. 
This  curve  has  been  called  the  "Helen  of  geometers,"  on  account 

'  OruTrrs  de  Descartes  (Tannery  et  Adam),  1897,  I,  400;  TT,  316.  See  also  G. 
Luri.i.  Ebcnc  Ciirvcn  (F.  Schiilte),  I,  1910,  p.  54. 


178  A  HISTORY  OF  MATHEMATICS 

of  its  beautiful  properties  and  the  controversies  which  their  discovery 
occasioned.  Its  quadrature  by  Roberval  was  generally  considered  a 
brilliant  achievement,  but  Descartes  commented  on  it  by  saying  that 
any  one  moderately  well  versed  in  geometry  might  have  done  this. 
He  then  sent  a  short  demonstration  of  his  own.  On  Roberval's  in- 
timating that  he  had  been  assisted  by  a  knowledge  of  the  solution, 
Descartes  constructed  the  tangent  to  the  curve,  and  challenged 
Roberval  and  Fermat  to  do  the  same.  Fermat  accomplished  it,  but 
Roberval  never  succeeded  in  solving  this  problem,  which  had  cost 
the  genius  of  Descartes  but  a  moderate  degree  of  attention. 

The  application  of  algebra  to  the  doctrine  of  curved  Hues  reacted 
favorably  upon  algebra.  As  an  abstract  science,  Descartes  improved 
it  by  the  introduction  of  the  modern  exponential  notation.  In  his 
Geomelrie,  1637,  he  writes  '' aa  ou  a~  pour  multiplier  a  par  soimenie; 
et  a^  pour  le  multiplier  encore  une  fois  par  a,  et  ainsi  a  I'iniini." 
Thus,  while  F.  Vieta  represented  A^  by  "A  cubus"  and  Stevin  x^ 
by  a  figure  3  within  a  small  circle,  Descartes  wrote  a^.  In  his  GeomeUie 
he  does  not  use  negative  and  fractional  exponents,  nor  literal  ex- 
ponents. His  notation  was  the  outgrowth  and  an  improvement  of 
notations  employed  by  writers  before  him.  Nicolas  Chuquet's  manu- 
script work,  Le  Tfiparty  en  la  science  des  nombres,^  1484,  gives  t2x^ 
and  io.v\  and  their  product  i2o.\;^  by  the  symbols  12^,  10^,  120^ 
respectively.  Chuquet  goes  even  further  and  writes  i2x°  and  fx~^ 
thus  12°,  7^™;  he  represents  the  product  of  8x^  and  ']X~'^  by  56^  J. 
Biirgi,  Reymer  and  J.  Kepler  use  Roman  numerals  for  the  exponen- 
tial symbol.    J.  Biirgi  writes  idx"^  thus  ^\.    Thomas  Harriot  simply 

repeats  the  letters;  he  writes  in  his  Artis  analytics  praxis  (1631), 
a^— io24a-+6254a,  thus:  aaaa— io2Ataa+()2$^a. 

Descartes'  exponential  notation  spread  rapidly;  about  1660  or 
1670  the  positive  integral  exponent  had  won  an  undisputed  place  in 
algebraic  notation.  In  1656  J.  Wallis  speaks  of  negative  and  fractional 
"indices,"  in  his  Arithmelica  infinitorum,  but  he  does  not  actually 

write  a~^  for  ~,  or  a'/'  for  ^/a^.    It  was  I.  Newton  who,  in  his  famous 

letter  to  H.  Oldenburg,  dated  June  13,  1676,  and  containing  his  an- 
nouncement of  the  binomial  theorem,  first  uses  negative  and  fractional 
exponents. 

With  Descartes  a  letter  represented  always  only  a  positive  number. 
It  was  Johann  Hudde  who  in  1659  first  let  a  letter  stand  for  negative 
as  well  as  positive  values. 

Descartes  also  established  some  theorems  on  the  theory  of  equa- 
tions.   Celebrated  is  his  "rule  of  signs"  for  determining  the  number 

1  Cluiqiiet's  "Le  Triparty,"  Bullettino  Boncompagni,  Vol.  13,  1880,  p.  740. 


DESCARTES  TO  NEWTON  179 

of  positive  and  negative  roots.  He  gives  the  rule  after  pointing  out 
the  roots  2,  3,  4,  —5  and  the  corresponding  binomial  factors  of  the 
equation  .T^—4.r^—igx'-+T:o6x— 120  =  0.  His  exact  words  are  as 
follows: 

"On  connoit  aussi  de  ceci  combien  il  pent  y  avoir  de  vraies  racines 
et  combien  de  fausses  en  chaque  equation:  a  savoir  il  y  en  pent  avoir 
autant  de  vraies  que  les  signes  +  et  —  s'y  trouvent  de  fois  etre 
changes,  et  autant  de  fausses  qu'il  s'y  trouve  de  fois  deux  signes  + 
ou  deux  signes  —  qui  s'entre-suivent.  Comme  en  la  derniere,  a  cause 
qu'apres  +x^  il  y  a  —  4.v'^,  qui  est  un  changement  du  signe  +  en  — , 
et  apres  —  ig.v'-  il  y  a  +io6x,  et  apres  +106.V  il  y  a  —120,  qui  sont 
encore  deux  aulres  changements,  ou  connoit  qu'il  y  a  trois  vraies 
lacines;  et  une  fausse,  a  cause  que  les  deux  signes  —  de  4.1-^  et  19.^'^ 
s'entre-suivent." 

This  statement  lacks  completeness.  For  this  reason  he  has  been 
frequently  criticized.  J.  Wallis  claimed  that  Descartes  failed  to 
notice  that  the  rule  breaks  down  in  case  of  imaginary  roots,  but 
Descartes  does  not  say  that  the  equation  always  has,  but  that  it  may 
have,  so  many  roots.  Did  Descartes  receive  any  suggestion  of  his 
rule  from  earlier  writers?  He  might  have  received  a  hint  from  H. 
Cardan,  whose  remarks  on  this  subject  have  been  summarized  by 
G.  Enestrom  '  as  follows:  If  in  an  equation  of  the  second,  third  or 
fourth  degree,  (i)  the  last  term  is  negative,  then  one  variation  of  sign 
signifies  one  and  only  one  positive  root,  (2)  the  last  term  is  positive, 
then  two  variations  indicate  either  several  positive  roots  or  none. 
Cardan  does  not  consider  equations  having  more  than  two  variations. 
G.  W.  Leibniz  was  the  first  to  erroneously  attribute  the  rule  of  signs 
to  T.  Harriot.  Descartes  was  charged  by  J.  Wallis  with  availing 
himself,  without  acknowledgment,  of  Harriot's  theory  of  equations, 
particularly  his  mode  of  generating  equations;  but  there  seems  to  be 
no  good  ground  for  the  charge. 

In  mechanics,  Descartes  can  hardly  be  said  to  have  advanced  be- 
yond Galileo.  The  latter  had  overthrown  the  ideas  of  Aristotle  on 
this  subject,  and  Descartes  simply  "threw  himself  upon  the  enemy" 
that  had  already  been  "put  to  the  rout."  His  statement  of  the  first 
and  second  laws  of  motion  was  an  improvement  in  form,  but  his  third 
law  is  false  in  substance.  The  motions  of  bodies  in  their  direct  impact 
was  imperfectly  understood  by  Galileo,  erroneously  given  by  Descartes, 
and  first  correctly  stated  by  C.  Wren,  J.  Wallis,  and  C.  Huygens. 

One  of  the  most  devoted  pupils  of  Descartes  was  the  learned 
Princess  Elizabeth,  daughter  of  Frederick  V.  She  applied  the  new 
analytical  geometry  to  the  solution  of  the  "Apollonian  problem." 
His  second  royal  follower  was  Queen  Christina,  the  daughter  of  Gus- 
tavus  Adolphus.  She  urged  upon  Descartes  to  come  to  the  Swedish 
court.  After  much  hesitation  he  accepted  the  invitation  in  1649. 
'  Biblioiheca  matliemalica,  3rd  S.,  Vol.  7,  1906-7,  p.  293. 


i8o  A  HISTORY  OF  MATHEMATICS 

He  died  at  Stockholm  one  year  later.  His  life  had  been  one  long  war- 
fare against  the  prejudices  of  men. 

It  is  most  remarkable  that  the  mathematics  and  philosophy  of 
Descartes  should  at  first  have  been  appreciated  less  by  his  country- 
men than  by  foreigners.  The  indiscreet  temper  of  Descartes  alienated 
the  great  contemporary  French  mathematicians,  Roberval,  Fermat, 
Pascal.  They  continued  in  investigations  of  their  own,  and  on  some 
points  strongly  opposed  Descartes.  The  universities  of  France  were 
under  strict  ecclesiastical  control  and  did  nothing  to  introduce  his 
mathematics  and  philosophy.  It  was  in  the  youthful  universities  of 
Holland  that  the  effect  of  Cartesian  teachings  was  most  immediate 
and  strongest. 

The  only  prominent  Frenchman  who  immediately  followed  in  the 
footsteps  of  the  great  master  was  Florimond  de  Beaune  (1601-1652). 
He  was  one  of  the  first  to  point  out  that  the  properties  of  a  curve 
can  be  deduced  from  the  properties  of  its  tangent.  This  mode  of 
inquiry  has  been  called  the  inverse  method  of  tangents.  He  contributed 
to  the  theory  of  equations  by  considering  for  the  first  time  the  upper 
and  lower  limits  of  the  roots  of  numerical  equations. 

In  the  Netherlands  a  large  number  of  distinguished  mathematicians 
were  at  once  struck  with  admiration  for  the  Cartesian  geometry. 
Foremost  among  these  are  van  Schooten,  John  de  Witt,  van  Heuraet, 
Sluze,  and  Hudde.  Franciscus  van  Schooten  (died  1660),  professor 
of  mathematics  at  Leyden,  brought  out  an  edition  of  Descartes' 
geometry,  together  with  the  notes  thereon  by  De  Beaune.  His  chief 
work  is  his  Excrcitationes  MathematiccB,  1657,  in  which  he  applies  the 
analytical  geometry  to  the  solution  of  many  interesting  and  difficult 
problems.  The  noble-hearted  Johann  de  Witt  (1625-1672),  grand- 
pensioner  of  Holland,  celebrated  as  a  statesman  and  for  his  tragical 
end,  was  an  ardent  geometrician.  He  conceived  a  new  and  ingenious 
way  of  generating  conies,  which  is  essentially  the  same  as  that  by 
projective  pencils  of  rays  in  modern  synthetic  geometry.  He  treated 
the  subject  not  synthetically,  but  with  aid  of  the  Cartesian  analysis. 
Rene  Frangois  de  Sluse  (1622-1685)  and  Johann  Hudde  (1633- 
1704)  made  some  improvements  on  Descartes'  and  Fermat's  methods 
of  drawing  tangents,  and  on  the  theory  of  maxima  and  minima.  With 
Hudde,  we  find  the  first  use  of  three  variables  in  analytical  geometry. 
He  is  the  author  of  an  ingenious  rule  for  finding  equal  roots.  We 
illustrate  it  by  the  equation  x^— x^— 8:c-f  12=0.  Taking  an  arith- 
metical progression  3,  2,  i,  o,  of  which  the  highest  term  is  equal  to 
the  degree  of  the  equation,  and  multiplying  each  term  of  the  equation 
respectively  by  the  corresponding  term  of  the  progression,  we  get 
3.r^-2x^— 8x=o,  or  3^-— 2.v— 8  =  0.  This  last  equation  is  by  one 
degree  lower  than  the  original  one.  Find  the  G.C.D.  of  the  two 
equations.  This  is  .r— 2;  hence  2  is  one  of  the  two  equal  roots.  Had 
there  been  no  common  divisor,  then  the  original  equation  would  no^ 


DESCARTES  TO  NEWTOK  i8i 

have  possessed  equal  roots.  Hudde  gave  a  demonstration  for  this 
rule.i 

Heinrich  van  Heuraet  must  be  mentioned  as  one  of  the  earUest 
geometers  who  occupied  themselves  with  success  in  the  rectification 
of  carves.  He  observed  in  a  general  way  that  the  two  problems  of 
quadrature  and  of  rectification  are  really  identical,  and  that  the  one 
can  be  reduced  to  the  other.  Thus  he  carried  the  rectification  of  the 
hyperbola  back  to  the  quadrature  of  the  hyperbola.  The  curve  which 
John  Walli.s  named  the  "semi-cubical  parabola,"  y^=ax-,  was  the 
first  curve  to  be  rectified  absolutely.  This  appears  to  have  been 
accompliihed  independently  by  P.  Fermat  in  France,  Van  Heuraet 
in  Holland  and  by  William  Neil  (1637-1670)  in  England.  According 
to  J.  Waliis  the  priority  belongs  to  Neil.  Soon  after,  the  cycloid  was 
rectified  by  C.  Wren  and  Fermat. 

A  mathematician  of  no  mean  ability  was  Gregory  St.  Vincent 
(15S4-1667),  a  Belgian,  who  studied  under  C.  Clavius  in  Rome  and 
was  two  years  professor  at  Prague,  where,  during  war  time,  his  manu- 
script volume  on  geometry  and  statics  was  lost  in  a  fire.  Other  papers 
of  his  were  saved  but  carried  about  for  ten  years  before  they  came 
again  into  his  possession,  at  his  home  in  Ghent.  They  became  the 
groundwork  of  his  great  book,  the  Opus  geometricum  quadratures 
circuli  et  sedionum  coni,  Antwerp,  1647.  It  consists  of  1225  folio 
pages,  divided  into  ten  books,  St.  Vincent  proposes  four  methods  for 
squaring  the  circle,  but  does  not  actually  carry  them  out.  The  work 
was  attacked  by  R.  Descartes,  M.  Mersenne  and  G.  P.  Roberval, 
and  defended  by  the  Jesuit  Alfons  Anton  de  Sarasa  and  others. 
Though  erroneous  on  the  possibility  of  squaring  the  circle,  the  Opus 
contains  solid  achievements,  which  were  the  more  remarkable,  because 
at  that  time  only  four  of  the  seven  books  of  the  conies  of  Apollonius 
of  Perga  were  known  in  the  Occident.  St.  Vincent  deals  with  conies, 
surfaces  and  solids  from  a  new  point  of  view,  employing  infinitesimals 
in  a  way  perhaps  less  objectionable  than  in  B.  Cavalieri's  book.  St. 
Vincent  was  probably  the  first  to  use  the  word  exhaurire  in  a  geo- 
metrical sense.  From  this  word  arose  the  name  of  "method  of  ex- 
haustion," as  applied  to  the  method  of  Euclid  and  Archimedes.  St. 
\'incent  used  a  method  of  transformation  of  one  conic  into  another, 
called  per  subtendas  (by  chords),  which  contains  germs  of  analytic 
geometry.  He  created  another  special  method  which  he  called  Ductus 
plant  in  planum  and  used  in  the  study  of  solids."  Unlike  Archimedes 
who  kept  on  dividing  distances,  only  until  a  certain  degree  of  small- 
ness  was  reached,  St.  Vincent  permitted  the  subdivisions  to  continue 

1  Heinrich  Suter,  Geschichte  dcr  Mathematischen  Wissenschaflen  Zurich,  2.  Theil, 
1875,  P-  25- 

-  See  M.  Marie,  Ilisloire  dcs  sciences  math.,  Vol.  3,  1884,  pp.  186-193;  Kari  Bopp, 
Kcgelschnilte  des  Gregorins  a  St.  Vinccnlo  in  Abhaiidl.  z.  Gesch.  d.  math.  Wissensch., 
XX  Heft,  1907,  pp.  83-314. 


i82  A  HISTORY  OF  MATHEMATICS 

ad  infinitum  and  obtained  a  geometric  series  that  was  infinite.  How- 
ever, infinite  series  had  been  obtained  before  him  by  Alvarus  Thomas, 
a  native  of  Lisbon,  in  a  work.  Liber  de  triplici  motii,  1509,^  and  possibly 
by  others.  But  St.  Vincent  was  the  first  to  apply  geometric  series  to 
the  "Achilles"  and  to  look  upon  the  paradox  as  a  question  in  the 
summation  of  an  infinite  series.  Moreover,  St.  Vincent  was  the  first 
to  state  the  exact  time  and  place  of  overtaking  the  tortoise.  He 
spoke  of  the  limit  as  an  obstacle  against  further  advance,  similar  to 
a  rigid  wall.  Apparently  he  was  not  troubled  by  the  fact  that  in  his 
theory,  the  variable  does  not  reach  its  limit.  His  exposition  of  the 
"Achilles"  was  favorably  received  by  G.  W.  Leibniz  and  by  writers 
over  a  century  afterward.  The  fullest  account  and  discussion  of 
Zeno's  arguments  on  motion  that  was  published  before  the  nineteenth 
century  was  given  by  the  noted  French  skeptical  philosopher,  Pierre 
Bayle,  in  an  article  "Zenon  d'Elee"  in  his  Dictionnaire  historique  et 
critique,  1696.^ 

The  prince  of  philosophers  in  Holland,  and  one  of  the  greatest 
scientists  of  the  seventeenth  century,  was  Christian  Huygens  (1629- 
1695),  a  native  of  The  Hague.  Eminent  as  a  physicist  and  astronomer, 
as  well  as  mathematician,  he  was  a  worthy  predecessor  of  Sir  Isaac 
Newton.  He  studied  at  Leyden  under  Frans  Van  Schooten.  The 
perusal  of  some  of  his  earhest  theorems  led  R.  Descartes  to  predict 
his  future  greatness.  In  1651  Huygens  wrote  a  treatise  in  which  he 
pointed  out  the  fallacies  of  Gregory  St.  Vincent  on  the  subject  of 
quadratures.  He  himself  gave  a  remarkably  close  and  convenient 
approximation  to  the  length  of  a  circular  arc.  In  1660  and  1663  he 
went  to  Paris  and  to  London.  In  1666  he  was  appointed  by  Louis 
XIV  member  of  the  French  Academy  of  Sciences.  He  was  induced 
to  remain  in  Paris  from  that  time  until  1681,  when  he  returned  to  his 
native  city,  partly  for  consideration  of  his  health  and  partly  on  ac- 
count of  the  revocation  of  the  Edict  of  Nantes. 

The  majority  of  his  profound  discoveries  were  made  with  aid  of  the 
ancient  geometry,  though  at  times  he  used  the  geometry  of  R.  Des- 
cartes or  of  B.  Cavalieri  and  P.  Fermat.  Thus,  like  his  illustrious 
friend.  Sir  Isaac  Newton,  he  always  showed  partiality  for  the  Greek 
geometry.  Newton  and  Huygens  were  kindred  minds,  and  had  the 
greatest  admiration  for  each  other.  Newton  always  speaks  of  him 
as  the  "Summus  Hugenius." 

To  the  two  curves  (cubical  parabola  and  cycloid)  previously  recti- 
fied he  added  a  third, — the  cissoid.  A  French  physician,  Claudius 
Perrault,  proposed  the  question,  to  determine  the  path  in  a  fixed  plane 
of  a  heavy  point  attached  to  one  end  of  a  taut  string  whose  other  end 
moves  along  a  straight  line  in  that  plane.  Huygens  and  G.  W.  Leibniz 
studied  this  problem  in  1693,  generalized  it,  and  thus  worked  out  the 

^  H.  Wieleitner,  in  Bibliolhcca  mathrmatica,  3.  F.,  Bd.  1914,  14,  p.  15:. 
^See  F.  Cajori  in  Am.  Math.  Monthly,  Vol.  22,  1915,  pp.  109-112. 


DESCARTES  TO  NEWTON  183 

geometry  of  the  "tractrix."  ^  Huygens  solved  the  problem  cf  the 
catenary,  determined  the  surface  of  the  parabolic  and  hyperbolic 
conoid,  and  discovered  the  properties  of  the  logarithmic  curve  and 
the  solids  generated  by  it.  Huygens'  De  horologio  oscillatorio  (Paris, 
1673)  is  a  work  that  ranks  second  only  to  the  Principia  of  Newton 
and  constitutes  historically  a  necessary  introduction  to  it.  The  book 
o{)ens  with  a  description  of  pendulum  clocks,  of  which  Huygens  is  the 
iu\'entor.  Then  follows  a  treatment  of  accelerated  motion  of  bodies 
falling  free,  or  sliding  on  inclined  planes,  or  on  given  curves, — cul- 
minating in  the  brilliant  discovery  that  the  cycloid  is  the  tautochronous 
curve.  To  the  theory  of  curves  he  added  the  important  theory  of 
"evolutes."  After  explaining  that  the  tangent  of  the  evolute  is 
normal  to  the  involute,  he  applied  the  theory  to  the  cycloid,  and 
showed  by  simple  reasoning  that  the  evolute  of  this  curve  is  an  equal 
cycloid.  Then  comes  the  complete  general  discussion  of  the  centre 
of  oscillation.  This  subject  had  been  proposed  for  investigation  by 
M.  JNIersenne  and  discussed  by  R.  Descartes  and  G.  P.  Roberval. 
In  Huygens'  assumption  that  the  common  centre  of  gravity  of  a 
group  of  bodies,  oscillating  about  a  horizontal  axis,  rises  to  its  original 
height,  but  no  higher,  is  expressed  for  the  first  time  one  of  the  most 
beautiful  principles  of  dynamics,  afterwards  called  the  principle  of 
the  conservation  of  vis  viva.  The  thirteen  theorems  at  the  close  of 
the  work  relate  to  the  theory  of  centrifugal  force  in  circular  motion. 
This  theory  aided  Newton  in  discovering  the  law  of  gravitation.  - 

Huygens  wrote  the  first  formal  treatise  on  probabihty.  He  pro- 
posed the  wave-theory  of  light  and  with  great  skill  applied  geometry 
to  its  development.  This  theory  was  long  neglected,  but  was  revived 
and  elaborated  by  Thomas  Young  and  A.  J.  Fresnel  a  century  later. 
Huygens  and  his  brother  improved  the  telescope  by  devising  a  better 
way  of  grinding  and  polishing  lenses.  With  more  efficient  instru- 
ments he  determined  the  nature  of  Saturn's  appendage  and  solved 
other  astronomical  questions.  Huygens'  Opuscula  posthuma  appeared 
in  1703. 

The  theory  of  combinations,  the  primitive  notions  of  which  go 
back  to  ancient  Greece,  received  the  attention  of  William  Buckley 
of  King's  College,  Cambridge  (died  1550),  and  especially  of  Blaise 
Pascal  who  treats  of  it  in  his  Arithmetical  Triangle.  Before  Pascal, 
this  Triangle  had  been  constructed  by  N.  Tartaglia  and  M.  Stifel. 
Fermat  applied  combinations  to  the  study  of  probabihty.  The  earliest 
mathematical  work  of  Leibniz  was  his  De  arte  combinatoria.  The 
subject  was  treated  by  John  Wallis  in  his  Algebra. 

John  Wallis  (1616-1703)  was  one  of  the  most  original  mathemati- 
cians of  his  day.  He  was  educated  for  the  Church  at  Cambridge  and  en- 

•  G.  Loria,  Rbene  Curven  (F.  Schiitte)  II,  iqii,  p.  188. 

-  E.  DiihrinR,  Kritischc  Geschichte  der  Allgetneinen  Principien  der  Mechanik. 
Leipzig,  1887,  p.  135. 


i84  A  HISTORY  OF  MATHEMATICS 

tered  Holy  Orders.  But  his  genius  was  employed  chiefly  in  the  study  of 
mathematics.  In  1649  he  was  appointed  Savilian  professor  of  geometry 
at  Oxford.  He  was  one  of  the  original  members  of  the  Royal  Society, 
which  was  founded  in  1663.  He  ranks  as  one  of  the  world's  greatest  de- 
cipherers of  cryptic  writing.^  Wallis  thoroughly  grasped  the  mathemat- 
ical methods  both  of  B.  Cavalieri  and  R.  Descartes.  His  Conic  Sections 
is  the  earliest  work  in  which  these  curves  are  no  longer  considered  as 
sections  of  a  cone,  but  as  curves  of  the  second  degree,  and  are  treated 
analytically  by  the  Cartesian  method  of  co-ordinates.  In  this  work 
Waliis  speaks  of  Descartes  in  the  highest  terms,  but  in  his  Algebra 
(1685,  Latin  edition  1693),  he,  without  good  reason,  accuses  Descartes 
of  plagiarizing  from  T.  Harriot.  It  is  interesting  to  observe  that,  in 
his  Algebra,  Wallis  discusses  the  possibility  of  a  fourth  dimension. 
Whereas  nature,  says  Wallis,  "doth  not  admit  of  more  than  three 
(local)  dimensions  ...  it  may  justly  seem  very  improper  to  talk  of 
a  solid  .  .  .  drawn  into  a  fourth,  fifth,  sixth,  or  further  dimension.  .  .. 
Nor  can  our  fansie  imagine  how  there  should  be  a  fourth  local  dimen- 
sion beyond  these  three."  ^  The  first  to  busy  himself  with  the  number 
of  dimensions  of  space  was  Ptolemy.  Wallis  felt  the  need  of  a  method 
of  representing  imaginaries  graphically,  but  he  failed  to  discover  a 
general  and  consistent  representation.^  He  published  Nasir-Eddin's 
proof  of  the  parallel  postulate  and,  abandoning  the  idea  of  equi- 
distance that  had  been  employed  without  success  by  F.  Commandino, 
C.  S.  Clavio,  P.  A.  Cataldi  and  G.  A.  Borelli,  gave  a  proof  of  his  own 
based  on  the  axiom  that,  to  every  figure  there  exists  a  similar  figure 
of  arbitrary  magnitude.^  The  existence  of  similar  triangles  was  as- 
sumed 1000  years  before  Wallis  by  Aganis,  who  was  probably  a 
teacher  of  Simplicius.  We  have  already  mentioned  elsewhere  Wallis's 
solution  of  the  prize  questions  on  the  cycloid,  which  were  proposed  by 
Pascal. 

The  Arithmetica  infinitorum,  published  in  1655,  is  his  greatest  work. 
By  the  application  of  analysis  to  the  Method  of  Indivisibles,  he  greatly 
increased  the  power  of  this  instrument  for  effecting  quadratures.  He 
created  the  arithmetical  conception  of  a  limit  by  considering  the 
successive  values  of  a  fraction,  formed  in  the  study  of  certain  ratios; 
these  fractional  values  steadily  approach  a  limiting  value,  so  that 
the  difference  becomes  less  than  any  assignable  one  and  vanishes 
when  the  process  is  carried  to  infinity.  He  advanced  beyond  J.  Kepler 
by  making  more  extended  use  of  the  "law  of  continuity"  and  placing 

1  D.  E.  Smith  in  Bull.  Am.  Math.  Soc,  Vol.  24,  1917,  p.  82. 

2  G.  Enestrom  in  Bibliotheca  mathematica,  3.  S.,  Vol.  12,  1911-12,  p.  88. 

3  See  Wallis'  Algebra,  1685,  pp.  264-273;  see  also  Enestrom  in  Bibliotheca  mathe- 
matica, 3.  S.,  Vol.  7,  pp.  263-269. 

"R.  Bonola,  op.  cit.,  pp.  12-17.  See  also  F.  Engel  u.  P.  Stackel,  Theorte  der 
ParalleUinien  von  Euclid  bis  auf  Gaiiss,  Leipzig,  1895,  pp.  21-36.  This  treatise 
gives  translations  into  German  of  Saccheri,  also  the  essays  of  Lambert  and  Taurinus, 
and  letters  of  Gauss. 


DESCARTES  TO  NEWTON  185 

full  reliance  in  it.  By  this  law  he  was  led  to  regard  the  denominators 
of  fractions  as  powers  with  negative  exponents.  Thus,  the  descending 
geometrical  progression  x^,  x',  x^,  x^,  if  continued,  gives  .x"^,  x~',  x~^, 

etc.;  which  is  the  same  thing  as  -,  -^,  -^.  The  exponents  of  this  geo- 
metric series  are  in  continued  arithmetical  progression,  3,  2,  i,  o, 

—  I,  —2,  —3.  However,  Wallis  does  not  actually  use  here  the  no- 
tation x~^,  :v~^,  etc.;  he  merely  speaks  of  negative  exponents.  He 
also  used  fractional  exponents,  which,  like  the  negative,  had  been 
invented  long  before,  but  had  failed  to  be  generally  introduced.  The 
symbol  00  for  infinity  is  due  to  him.  Wallis  introduces  the  name, 
" h\T)ergeometric  series"  for  a  series  different  from  a,  ab,  ab^,  .  .  .  ; 
he  did  not  look  upon  this  new  series  as  a  power-series  nor  as  a  function 
of  .V. 

B.  Cavalieri  and  the  French  geometers  had  ascertained  the  formula 
for  squaring  the  parabola  of  any  degree,  y=.r'",  tn  being  a  positive 
integer.  By  the  summation  of  the  powers  of  the  terms  of  infinite 
arithmetical  series,  it  was  found  that  the  curve  y=x"'  is  to  the  area 
of  the  parallelogram  having  the  same  base  and  altitude  as  i  is  to 
m+i.  Aided  by  the  law  of  continuity,  Wallis  arrived  at  the  result 
that  this  formula  holds  true  not  only  when  m  is  positive  and  integral, 
but  also  when  it  is  fractional  or  negative.  Thus,  in  the  parabola 
y=\/px,  m  =  \\  hence  the  area  of  the  parabolic  segment  is  to  that 
of  the  circumscribed  rectangle  as  i  :  i|,  or  as  2:3.  Again,  suppose 
that  in  y  =  .v'",  w=— f ;  then  the  curve  is  a  kind  of  hyperbola  referred 
to  its  asymptotes,  and  the  hyperbolic  space  between  the  curve  and 
its  asymptotes  is  to  the  corresponding  parallelogram  as  i  :  ^.     If  m  = 

—  I,  as  in  the  common  equilateral  hj^jerbola  y=x~^  or  xy=i,  then 
this  ratio  is  i  :  —  i  +  i,  or  1:0,  showing  that  its  asymptotic  space 
is  infinite.  But  in  the  case  when  m  is  greater  than  unity  and  negative, 
Wallis  was  unable  to  interpret  correctly  his  results.  For  example, 
if  w=— 3,  then  the  ratio  becomes  i  :  —  2,  or  as  unity  to  a  negative 
number.  What  is  the  meaning  of  this?  Wallis  reasoned  thus:  If  the 
denominator  is  only  zero,  then  the  area  is  already  infinite;  but  if  it  is 
less  than  zero,  then  the  area  must  be  more  than  infinite.  It  was 
pointed  out  later  by  P.  Varignon,  that  this  space,  supposed  to  exceed 
infinity,  is  really  finite,  but  taken  negatively;  that  is,  measured  in  a 
contrary  direction.^    The  method  of  Wallis  was  easily  extended  to 

m  p 

cases  such  as  y^axn+bxi  by  performing  the  quadrature  for  each  term 
separately,  and  then  adding  the  results. 

The  manner  in  which  Wallis  studied  the  quadrature  of  the  circle 
and  arrived  at  his  expression  for  the  value  of  7r  is  extraordinary.  He 
found  that  the  areas  comprised  between  the  axes,  the  ordinate  cor- 

'  J.  F.  Montucla,  Histoire  dcs  mathematipues.  Paris.  Tome  2,  An  VII,  p.  .^"^o. 


i86  A  HISTORY  OF  MATHEMATICS 

responding  to  x,  and  the  curves  represented  by  the  equations  y= 
{i-x^f,  y^ii-x^y,  y^{i-x'^y,  y={i-x~Y^  etc.,  are  expressed  ir 
functions  of  the  circumscribed  rectangles  having  x  and  y  for  theii 
sides,  by  the  quantities  forming  the  series 

X, 

x-^x'^, 

x-fx^+fx^— ^x^,  etc. 
When  x=i,  these  values  become  respectively  i,  |,  1^5-,  iV?,  etc.  Now 
since  the  ordinate  of  the  circle  is  y=(i-x2)^,  the  exponent  of  which  is 
I  or  the  mean  value  between  o  and  i,  the  question  of  this  quadrature 
reduced  itself  to  this:  If  o,  i,  2,  3,  etc.,  operated  upon  by  a  certain  law, 
give  I,  I,  A,  1V5.  what  will  \  give,  when  operated  upon  by  the  same 
law?  He  attempted  to  solve  this  by  interpolation,  a  method  first 
brought  into  prominence  by  him,  and  arrived  by  a  highly  complicated 
and  difficult  analysis  at  the  following  very  remarkable  expression: 

TT     2  .2. 4.4. 6. 6. 8. 8.  ■  ■ 

"2~i.3-3-5-5-7-7-9---' 
He  did  not  succeed  in  making  the  interpolation  itself,  because  he 
did  not  employ  literal  or  general  exponents,  and  could  not  conceive  a 
series  with  more  than  one  term  and  less  than  two,  which  it  seemed 
to  him  the  interpolated  series  must  have.  The  consideration  of  this 
difficulty  led  I.  Newton  to  the  discovery  of  the  Binomial  Theorem. 
This  is  the  best  place  to  speak  of  that  discovery.  Newton  virtually 
assumed  that  the  same  conditions  which  underhe  the  general  ex- 
pressions for  the  areas  given  above  must  also  hold  for  the  expression 
to  be  interpolated.  In  the  first  place,  he  observed  that  in  each  ex- 
pression the  first  term  is  x,  that  x  increases  in  odd  powers,  that  the 
signs  alternate  -t-  and  — ,  and  that  the  second  terms  fx^  il•^  |x^  fx^ 
are  in  arithmetical  progression.     Hence  the  first  two  terms  of  the 

interpolated  series  must  be  x-^— .  He  next  considered  that  the  de- 
nominators I,  3,  5,  7,  etc.,  are  in  arithmetical  progression,  and  that 
the  coefficients  in  the  numerators  in  each  expression  are  the  digits 
of  some  power  of  the  number  11;  namely,  for  the  first  expression,  ii*^ 
or  i;  for  the  second,  11^  or  i,  i;  for  the  third,  11^  or  i,  2,  i;  for  the 
fourth,  11^  or  I,  3,  3,  i;  etc.  He  then  discovered  that,  having  given 
the  second  digit  (call  it  m),  the  remaining  digits  can  be  found  by  con- 

.      m—o    m—i     m—2 
tinual  multipHcation  of  the  terms  of  the  series ,  —^  •  — —  • 

^^^  .  etc.     Thus,  if  w=4,  then  4.  ^^^— ^  gives  6;  6 .  — — -  gives  4; 
4  23 


DESCARTES  TO  NEWTON  187 

4  . ^  gives  I.     Applying  this  rule  to  the  required  series,  since  the 

4 

1^.3 
second  term  is  ^^-,  we  have  w  =  |,  and  then  get  for  the  succeeding 

3 

coefFicienls  in   the  numerators  respectively   —  I,   —  A,  ~i2  8"'  ^^c.; 

hence  the  required  area  for  the  circular  segment  is  x—— —  — 

etc.  Thus  he  found  the  interpolated  expression  to  be  an  infinite  series, 
instead  of  one  having  more  than  one  term  and  less  than  two,  as  Wallis 
believed  it  must  be.  This  interpolation  suggested  to  Newton  a  mode 
of  expanding  (i  — .v-)^,  or,  more  generally,  (i  — .v'-)'",  into  a  series.  He 
observed  that  he  had  only  to  omit  from  the  expression  just  found  the 
denominators  i,  3,  5,  7,  etc.,  and  to  lower  each  power  of  x  by  unity, 
and  he  had  the  desired  expression.  In  a  letter  to  H.  Oldenburg 
(June  13,  1676^  Newton  states  the  theorem  as  follows:  The  extraction 
of  roots  is  much  shortened  by  the  theorem 

n    ^       2n      ^       3W       ^- 

where  .1  means  the  first  term,  P'T,  B  the  second  term,  C  the  third 
term,  etc.  He  verified  it  by  actual  multiphcation,  but  gave  no  regular 
proof  of  it.  He  gave  it  for  any  exponent  whatever,  but  made  no  dis- 
tinction between  the  case  when  the  exponent  is  positive  and  integral, 
and  the  others. 

It  should  here  be  mentioned  that  very  rude  beginnings  of  the  bi- 
nomial theorem  are  found  very  early.  The  Hindus  and  Arabs  used 
the  expansions  of  (a+b)'^  and  (a+b)^  for  extracting  roots;  Vieta  knew 
the  expansion  of  (a+b)'^;  but  these  were  the  results  of  simple  multi- 
plication without  the  discovery  of  any  law.  The  binomial  coeflScients 
for  positive  whole  exponents  were  known  to  some  Arabic  and  Euro- 
pean mathematicians.  B.  Pascal  derived  the  coefficients  from  the 
method  of  what  is  called  the  "arithmetical  triangle."  Lucas  de 
Burgo,  M.  Stifel,  S.  Stevinus,  H.  Briggs,  and  others,  all  possessed 
something  from  which  one  would  think  the  binomial  theorem  could 
have  been  gotten  with  a  little  attention,  "if  we  did  not  know  that 
such  simple  relations  were  difficult  to  discover," 

Though  Wallis  had  obtained  an  entirely  new  expression  for  tt,  he 
was  not  satisfied  with  it;  for  instead  of  a  finite  number  of  terms  yield- 
ing an  absolute  value,  it  contained  an  infinite  number,  approaching 
nearer  and  nearer  to  that  value.  He  therefore  induced  his  friend,  Lord 
Brouncker,  the  first  president  of  the  Royal  Society,  to  investigate 
this  suliject.  Of  course  Lord  Brouncker  did  not  find  what  they  were 
after,  but  he  obtained  the  following  beautiful  equaUty: — 


i88  A  HISTORY  OF  MATHEMATICS 


2  +  - 


2  +  - 


25 


2+~       ^9 


2+  etc. 

Continued  fractions,  both  ascending  and  descending,  appear  to  have 
been  known  already  to  the  Greeks  and  Hindus,  though  not  in  our 
present  notation.  Brouncker's  expression  gave  birth  to  the  theory  of 
continued  fractions. 

Wallis'  method  of  quadratures  was  diUgently  studied  by  his  dis- 
ciples. Lord  Brouncker  obtained  the  first  infinite  series  for  the  area 
of  the  equilateral  hyperbola  ^17=1  between  one  of  its  asymptotes  and 

the  ordinates  for  a;=i  and  x=2;  vii:.  the  area 1 1-— 7+  •  •  •    The 

1.2    3.4    5.6 

Logarithmotechnia  (London,  1668)  of  Nicolaus  Mercator  is  often  said 
to  contain  the  series  log  (i+a)=a + ...  In  reality  it  con- 
tains the  numerical  values  of  the  first  few  terms  of  that  series,  tak- 
ing a  =.i,  also  a =.2 1.  He  adhered  to  the  mode  of  exposition  which 
favored  the  concrete  special  case  to  the  general  formula.  Wallis  was 
the  first  to  state  Mercator's  logarithmic  series  in  general  symbols. 
Mercator  deduced  his  results  from  the  grand  property  of  the  hyper- 
bola deduced  by  Gregory  St.  Vincent  in  Book  VII  of  his  Opus  geo- 
metricum,  Antwerp,  1647:  If  parallels  to  one  asymptote  are  drawn 
between  the  hyperbola  and  the  other  asymptote,  so  that  the  successive 
areas  of  the  mixtilinear  quadrilaterals  thus  formed  are  equal,  then 
the  lengths  of  the  parallels  form  a  geometric  progression.  Apparently 
the  first  writer  to  state  this  theorem  in  the  language  of  logarithms 
was  the  Belgian  Jesuit  Alfons  Anton  de  Sarasa,  who  defended  Gregory 
St.  Vincent  against  attacks  made  by  Mersenne.  Mercator  showed 
how  the  construction  of  logarithmic  tables  could  be  reduced  to  the 
quadrature  of  hyperbolic  spaces.  Following  up  some  suggestions  of 
Wallis,  William  Neil  succeeded  in  rectifying  the  cubical  parabola,  and 
C.  Wren  in  rectifying  any  cycloidal  arc.  Gregory  St.  Vincent,  in 
Part  X  of  his  Opus  describes  the  construction  of  certain  quartic  curves, 
often  called  virtual  parabolas  of  St.  Vincent,  one  of  which  has  a  shape 
much  like  a  lemniscate  and  in  Cartesian  co-ordinates  is  d'{y^—x^)=y*. 
Curves  of  this  type  are  mentioned  in  the  correspondence  of  C.  Huy- 
gens  with  R.  de  Sluse,  and  with  G.  W.  Leibniz. 

A  prominent  English  mathematician  and  contemporary  of  WalUs 
was  Isaac  Barrow  (1630-167 7).  He  was  professor  of  mathematics 
in  London,  and  then  in  Cambridge,  but  in  1669  he  resigned  his  chaii 


DESCARTES  TO  NEWTON 


189 


to  his  illustrious  pupil,  Isaac  Newton,  and  renounced  the  study  of 
mathematics  for  that  of  divinity.  As  a  mathematician,  he  is  most 
celebrated  for  his  method  of  tangents.  He  simplified  the  method  of 
P.  Fermat  by  introducing  two  infinitesimals  instead  of  one,  and  ap- 
proximated to  the  course  of  reasoning  afterwards  followed  by  Newton 
in  his  doctrine  on  Ultimate  Ratios.  The  following  books  are  Barrow's: 
Lediones  geometric^  (1670),  Lectiones  mathematiccB  (1683-1685). 

He  considered  the  infinitesimal  right  triangle  ABB'  having  for  its 
sides  the  difference  between  two  successive  ordinates,  the  distance 
between  them,  and  the  portion  of  the  curve  intercepted  by  them. 
This  triangle  is  similar  to  BPT,  formed  by  the  ordinate,  the  tangent, 
and  the  sub-tangent.  Hence,  if  we 
know  the  ratio  of  B'A  to  BA,  then 
we  know  the  ratio  of  the  ordinate 
and  the  sub-tangent,  and  the  tangent 
can  be  constructed  at  once.  For  any 
curve,  say  y~=px,  the  ratio  of  B'A  to 
BA  is  determined  from  its  equation 
as  follows:  If  x  receives  an  infinitesi- 
mal increment  PP'  =  e,  then  y  receives 
an  increment  B'A=a,  and  the  equation  for  the  ordinate  B'P'  becomes 
y--\-2ay-\-a'^=px+pe.  Since  y^  =  px,  we  get  2ay+a'^=pe;  neglecting 
higher  powers  of  the  infinitesimals,  we  have  2ay=pe,  which  gives 

a:e=p:  2y  =  p:  2\/px. 
But  a:  g=the  ordinate:  the  sub-tangent;  hence 

p:  2\fpx=  y/px:  sub- tangent, 
giving  2x  for  the  value  of  the  sub-tangent.    This  method  differs  from 
that  of  the  differential  calculus  chiefly  in  notation.     In  fact,  a  recent 
investigator  asserts,  "Isaac  Barrow  was  the  first  inventor  of  the  in- 
finitesimal calculus."  ^ 

Of  the  integrations  that  were  performed  before  the  Integral  Calculus 
was  invented,  one  of  the  most  difficult  grew  out  of  a  practical  problem 
of  navigation  in  connection  with  Gerardus  M creator'' s  map.  In  1599 
Edward  Wright  published  a  table  of  latitudes  giving  numbers  express- 
ing the  length  of  an  arc  of  the  nautical  meridian.  The  table  was  com- 
puted by  the  continued  addition  of  the  secants  of  i" ,  2",  3",  etc.    In 


modem  symbols  this  amounts  to  r  I  sec  ^  J  ^  =  r  log  tan  (90° —  6)  li- 
lt was  Henry  Bond  who  noticed  by  inspection  about  1645  that 
Wright's  table  was  a  table  of  logarithmic  tangents.  Actual  demon- 
strations of  this,  thereby  really  establishing  the  above  definite  integral, 
were  given  by  James  Gregory  in  1668,  Isaac  Barrow  in  1670,  John 

'  J.  M.  Child,  The  Geometrical  Lectures  of  Isaac  Barrow,  Chicago  and  London, 
^916,  preface. 


(go  A  HISTORY  OF  MATHEMATICS 

Wallis  in  1685,  and  Edmund  Halley  in  1698.^    James  Gregory  and 
Barrow  gave  also  the  integral  /    tsind  d  0  =  log  sec  6;  B.  Cavalieri  in 

1647  established  the  integral  of  I    x"  dx.    Similar  results  were  obtained 

by  E.  Torricelli,  Gregory  St.  Vincent,  P.  Fermat,  G.  P.  Roberval  and 
B.  Pascal.' 

Newton  to  Eider 

It  has  been  seen  that  in  France  prodigious  scientific  progress  was 
made  during  the  beginning  and  middle  of  the  seventeenth  century. 
The  toleration  which  marked  the  reign  of  Henry  IV  and  Louis  XIII 
was  accompanied  by  intense  intellectual  activity.  Extraordinary  con- 
fidence came  to  be  placed  in  the  power  of  the  human  mind.  The  bold 
intellectual  conquests  of  R.  Descartes,  P.  Fermat,  and  B.  Pascal  en- 
riched mathematics  with  imperishable  treasures.  During  the  early 
part  of  the  reign  of  Louis  XIV  we  behold  the  sunset  splendor  of  this 
glorious  period.  Then  followed  a  night  of  mental  effeminacy.  This 
lack  of  great  scientific  thinkers  during  the  reign  of  Louis  XIV  may  be 
due  to  the  simple  fact  that  no  great  minds  were  born;  but,  according 
to  Buckle,  it  was  due  to  the  paternalism,  to  the  spirit  of  dependence 
and  subordination,  and  to  the  lack  of  toleration,  which  marked  the 
policy  of  Louis  XIV. 

In  the  absence  of  great  French  thinkers,  Louis  XIV  surrounded 
himself  by  eminent  foreigners.  O.  Romer  from  Denmark,  C.  Huygens 
from  Holland,  Dominic  Cassini  from  Italy,  were  the  mathematicians 
and  astronomers  adorning  his  court.  They  were  in  possession  of  a 
brilliant  reputation  before  going  to  Paris.  Simply  because  they  per- 
formed scientific  work  in  Paris,  that  work  belongs  no  more  to  France 
than  the  discoveries  of  R.  Descartes  belong  to  Holland,  or  those  of 
J.  Lagrange  to  Germany,  or  those  of  L.  Euler  and  J.  V.  Poncelet  to 
Russia.  We  must  look  to  other  countries  than  France  for  the  great 
scientific  men  of  the  latter  part  of  the  seventeenth  century. 

About  the  time  when  Louis  XIV  assumed  the  direction  of  the 
French  government  Charles  II  became  king  of  England.  At  this 
time  England  was  extending  her  commerce  and  navigation,  and  ad- 
vancing considerably  in  material  prosperity.  A  strong  intellectual 
movement  took  place,  which  was  unwittingly  supported  by  the  king. 
The  age  of  poetry  was  soon  followed  by  an  age  of  science  and  philos- 
ophy. In  two  successive  centuries  England  produced  Shakespeare 
and  I.  Newton! 

^  See  F.  Cajori  in  BibHothcca  mathemalica,  3.  S.,  Vol.  14,  1915.,  pp.  312-319. 
2H.  G.  Zeuthcn,  Gcschichtc  der  Math,  (deutsch  v.  R.  Meyer),  LeiDzig,  1903, 
pp.  256  ff. 


NEWTON  TO  EULER  19  r 

Germany  still  continued  in  a  state  of  national  degradation.  The 
Thirty  Years'  War  had  dismembered  the  empire  and  brutalized  the 
people.  Yet  this  darkest  period  of  Germany's  history  produced  G.  W. 
Leibniz,  one  of  the  greatest  geniuses  of  modern  times. 

There  are  certain  focal  points  in  history  toward  which  the  lines  of 
past  progress  converge,  and  from  which  radiate  the  advances  of  the 
future.  Such  was  the  age  of  Newton  and  Leibniz  in  the  history  of 
mathematics.  During  fifty  years  preceding  this  era  several  of  the 
brightest  and  acutest  mathematicians  bent  the  force  of  their  genius 
in  a  direction  which  finally  led  to  the  discovery  of  the  infinitesimal 
calculus  by  Newton  and  Leibniz.  B.  Cavalieri,  G.  P.  Roberval,  P. 
Fermat,  R.  Descartes,  J.  Wallis,  and  others  had  each  contributed  to 
the  new  geometry.  So  great  was  the  advance  made,  and  so  near  was 
their  approach  toward  the  invention  of  the  infinitesimal  analysis,  that 
both  J.  Lagrange  and  P.  S.  Laplace  pronounced  their  countryman, 
P.  Fermat,  to  be  the  first  inventor  of  it.  The  differential  calculus, 
therefore,  was  not  so  much  an  individual  discovery  as  the  grand  result 
of  a  succession  of  discoveries  by  different  minds.  Indeed,  no  great 
discovery  ever  flashed  upon  the  mind  at  once,  and  though  those  of 
Newton  will  influence  mankind  to  the  end  of  the  world,  yet  it  must  be 
admitted  that  Pope's  lines  are  only  a  "poetic  fancy": — 

"Nature  and  Nature's  laws  lay  hid  in  night; 
God  said,  'Let  Newton  be,'  and  all  was  light." 

Isaac  Newton  (1642-1727)  was  born  at  Woolsthorpe,  in  Lincoln- 
shire, the  same  year  in  which  Galileo  died.  At  his  birth  he  was  so 
small  and  weak  that  his  life  was  despaired  of.  His  mother  sent  him 
at  an  early  age  to  a  village  school,  and  in  his  twelfth  year  to  the  public 
school  at  Grantham.  At  first  he  seems  to  have  been  very  inattentive 
to  his  studies  and  very  low  in  the  school;  but  when,  one  day,  the  little 
Isaac  received  a  severe  kick  upon  his  stomach  from  a  boy  who  was 
above  him,  he  labored  hard  till  he  ranked  higher  in  school  than  his 
antagonist.  From  that  time  he  continued  to  rise  until  he  was  the 
head  boy.^  At  Grantham,  Isaac  showed  a  decided  taste  for  mechan- 
ical inventions.  Pie  constructed  a  water-clock,  a  wind-mill,  a  carriage 
moved  by  the  person  who  sat  in  it,  and  other  toys.  When  he  had  at- 
tained his  fifteenth  year  his  mother  took  hun  home  to  assist  her  in 
the  management  of  the  farm,  but  his  great  dislike  for  farmwork  and 
his  irresistible  passion  for  study,  induced  her  to  send  him  back  to 
Grantham,  where  he  remained  till  his  eighteenth  year,  when  he  en- 
tered Trinity  College,  Cambridge  (1660).  Cambridge  was  the  real 
birthplace  of  Newton's  genius.  Some  idea  of  his  strong  intuitive 
powers  may  be  drawn  from  the  fact  that  he  regarded  the  theorems  of 
ancient  geometry  as  self-evident  truths,  and  that,  without  any  pre- 
liminary study,  he  made  himself  master  of  Descartes'  Geometry.  He 
1  D.  Brewster,  The  Memoirs  of  Xeu'lon,  Edinburgh,  \'oI.  T,  1855,  p.  8. 


l92 


A  HISTORY  OF  MATHEMATICS 


afterwards  regarded  this  neglect  of  elementary  geometry  a  mistake 
in  his  mathematical  studies,  and  he  expressed  to  Dr.  H.  Pemberton 
his  regret  that  "he  had  applied  himself  to  the  works  of  Descartes  and 
other  algebraic  writers  before  he  had  considered  the  Elements  of  Euclid 
with  that  attention  which  so  excellent  a  writer  deserves."  Besides  R. 
Descartes'  Geometry,  he  studied  W.  Oughtred's  Clavis,  J.  Kepler's 
Optics,  the  works  of  F.  Vieta,  van  Schooten's  Miscellanies,  I.  Barrow's 
Lectures,  and  the  works  of  J.  Wallis.  He  was  particularly  deUghted 
with  Wallis'  Arithmetic  of  Infinites,  a  treatise  fraught  with  rich  and 
varied  suggestions.  Newton  had  the  good  fortune  of  having  for  a 
teacher  and  fast  friend  the  celebrated  Dr.  Barrow,  who  had  been 
elected  professor  of  Greek  in  1660,  and  was  made  Lucasian  professor 
of  mathematics  in  1663.  The  mathematics  of  Barrow  and  of  Wallis 
were  the  starting-points  from  which  Newton,  with  a  higher  power 
than  his  masters',  moved  onward  into  wider  fields.  Wallis  had  ef- 
fected the  quadrature  of  curves  whose  ordinates  are  expressed  by  any 
integral  and  positive  power  of  (i—x-).  We  have  seen  how  Wallis 
attempted  but  failed  to  interpolate  between  the  areas  thus  calculated, 
the  areas  of  other  curves,  such  as  that  of  the  circle;  how  Newton  at- 
tacked the  problem,  effected  the  interpolation,  and  discovered  the 
Binomial  Theorem,  which  afforded  a  much  easier  and  direct  access  to 
the  quadrature  of  curves  than  did  the  method  of  interpolation;  for 
even  though  the  binomial  expression  for  the  ordinate  be  raised  to  a 
fractional  or  negative  power,  the  binomial  could  at  once  be  expanded 
into  a  series,  and  the  quadrature  of  each  separate  term  of  that  series 
could  be  effected  by  the  method  of  Wallis.  Newton  introduced  the 
system  of  literal  indices. 

Newton's  study  of  quadratures  soon  led  him  to  another  and  most 
profound  invention.  He  himself  says  that  in  1665  and  1666  he  con- 
ceived the  method  of  fluxions  and  applied  them  to  the  quadrature  of 
curves.  Newton  did  not  communicate  the  invention  to  any  of  his 
friends  till  1669,  when  he  placed  in  the  hands  of  Barrow  a  tract,  en- 
titled De  Analysi  per  ALquationes  Numero  Terminoriim  Infinitas,  which 
was  sent  by  Barrow  to  John  Collins,  who  greatly  admired  it.  In 
this  treatise  the  principle  of  fluxions,  though  distinctly  pointed  out, 
is  only  partially  developed  and  explained.  Supposing  the  abscissa  to 
increase  uniformly  in  proportion  to  the  time,  he  looked  upon  the  area 
of  a  curve  as  a  nascent  quantity  increasing  by  continued  fluxion  in 
the  proportion  of  the  length  of  the  ordinate.  The  expression  which 
was  obtained  for  the  fluxion  he  expanded  into  a  finite  or  infinite  series 
of  monomial  terms,  to  which  Wallis'  rule  was  apphcable.  Barrow 
urged  Newton  to  publish  this  treatise;  "  but  the  modesty  of  the  author, 
of  which  the  excess,  if  not  culpable,  was  certainly  in  the  present  in- 
stance very  unfortunate,  prevented  his  compliance."  ^    Had  this  tract 

'John  Playfair,  "Progress  of  the  Mathematical  and  Physical  Sciences"  in  En- 
cydopadia  Britannica,  7  th  Edition. 


NEWTON  TO  EULER  19,? 

been  published  then,  instead  of  forty-two  years  later,  there  probably 
would  have  been  no  occasion  for  that  long  and  deplorable  controversy 
between  Newton  and  Leibniz. 

For  a  long  time  Newton's  method  remained  unknown,  except  to  his 
friends  and  their  correspondents.  In  a  letter  to  Collins,  dated  De- 
cember loth,  1672,  Newton  states  the  fact  of  his  invention  with  one 
example,  and  then  says:  "This  is  one  particular,  or  rather  corollary, 
of  a  general  method,  which  extends  itself,  without  any  troublesome 
calculation,  not  only  to  the  drawing  of  tangents  to  any  curve  lines, 
whether  geometrical  or  mechanical,  or  anyhow  respecting  right  lines 
or  other  curves,  but  also  to  the  resolving  other  abstruser  kinds  of 
problems  about  the  crookedness,  areas,  lengths,  centres  of  gravity  of 
cur\'es,  etc.;  nor  is  it  (as  Hudde's  method  of  Maximis  and  Minimis) 
limited  to  equations  which  are  free  from  surd  quantities.  This  method 
I  have  interwoven  with  that  other  of  working  in  equations,  by  reducing 
them  to  infinite  series." 

These  last  words  relate  to  a  treatise  he  composed  in  the  year  1671, 
entitled  Method  of  Fluxions,  in  which  he  aimed  to  represent  his  method 
as  an  independent  calculus  and  as  a  complete  system.  This  tract  was 
intended  as  an  introduction  to  an  edition  of  Kinckhuysen's  Algebra, 
which  he  had  undertaken  to  publish.  "But  the  fear  of  being  involved 
in  disputes  about  this  new  discovery,  or  perhaps  the  wish  to  render 
it  more  complete,  or  to  have  the  sole  advantage  of  employing  it  in  his 
physical  researches,  induced  him  to  abandon  this  design."  ^ 

Excepting  two  papers  on  optics,  all  of  his  works  appear  to  have 
been  published  only  after  the  most  pressing  solicitations  of  his  friends 
and  against  his  own  wishes.  His  researches  on  light  were  severely 
criticised,  and  he  wrote  in  1675:  "I  was  so  persecuted  with  discussions 
arising  out  of  my  theory  of  light  that  I  blamed  my  own  imprudence 
for  parting  with  so  substantial  a  blessing  as  my  quiet  to  run  after  a 
shadow." 

The  Method  of  Fluxions,  translated  by  J.  Colson  from  Newton's 
Latin,  was  first  published  in  1736,  or  sixty-five  years  after  it  was 
written.  In  it  he  explains  first  the  expansion  into  series  of  fractional 
and  irrational  quantities, — a  subject  which,  in  his  first  years  of  study, 
received  the  most  careful  attention.  He  then  proceeds  to  the  solution 
of  the  two  following  mechanical  problems,  which  constitute  the  pillars, 
so  to  speak,  of  the  abstract  calculus: — 

"I.  The  length  of  the  space  described  being  continually  (i.  e.  at 
all  times)  given;  to  find  the  velocity  of  the  motion  at  any  time  pro- 
posed. 

"II.  The  velocity  of  the  motion  being  continually  given;  to  find 
the  length  of  the  space  described  at  any  time  proposed." 

Preparatory  to  the  solution,  Newton  says:  "Thus,  in  the  equation 
y=x'^,  if  y  represents  the  length  of  the  space  at  any  time  described, 
>  D.  Brewster,  op.  cit.,  Vol.  2,  1855,  p.  15. 


194  A  HISTORY  OF  MATHEMATICS 

which  (time)  another  space  x,  by  increasing  with  an  uniform  celerity 
.V,  measures  and  exhibits  as  described:  then  2xx  will  represent  the 
celerity  by  which  the  space  y,  at  the  same  moment  of  time,  proceeds 
to  be  described;  and  contrarywise." 

"But  whereas  we  need  not  consider  the  time  here,  any  farther  than 
it  is  expounded  and  measured  by  an  equable  local  motion;  and  be- 
sides, whereas  only  quantities  of  the  same  kind  can  be  compared  to- 
gether, and  also  their  velocities  of  increase  and  decrease;  therefore,  in 
what  follows  I  shall  have  no  regard  to  time  formally  considered,  but 
I  shall  suppose  some  one  of  the  quantities  proposed,  being  of  the  same 
kind,  to  be  increased  by  an  equable  fluxion,  to  which  the  rest  may  be 
referred,  as  it  were  to  time;  and,  therefore,  by  way  of  analogy,  it 
may  not  improperly  receive  the  name  of  time."  In  this  statement  of 
Newton  there  is  contained  his  answer  to  the  objection  which  has  been 
raised  against  his  method,  that  it  introduces  into  analysis  the  foreign 
idea  of  motion.  A  quantity  thus  increasing  by  uniform  fluxion,  is 
what  we  now  call  an  independent  variable. 

Newton  continues:  "Now  those  quantities  which  I  consider  as 
gradually  and  indefinitely  increasing,  I  shall  hereafter  call  fluents,  or 
flowing  quantities,  and  shaU  represent  them  by  the  final  letters  of  the 
alphabet,  v,  x,  y,  and  z;  .  .  .  and  the  velocities  by  which  every  fluent 
is  increased  by  its  generating  motion  (which  I  may  call  fluxions,  or 
simply  velocities,  or  celerities),  I  shall  represent  by  the  same  letters 
pointed,  thus,  v,  x,  y,  z.  That  is,  for  the  celerity  of  the  quantity  v 
I  shall  put  V,  and  so  for  the  celerities  of  the  other  quantities  x,  y,  and 
2, 1  shall  put  X,  y,  and  z,  respectively."  It  must  here  be  observed  that 
Newton  does  not  take  the  fluxions  themselves  infinitely  small.  The 
"moments  of  fluxions,"  a  term  introduced  further  on,  are  infinitely 
smaU  quantities.  These  "moments,"  as  defined  and  used  in  the 
Method  of  Fluxions,  are  substantially  the  differentials  of  Leibniz.  De 
Morgan  points  out  that  no  small  amount  of  confusion  has  arisen  from 
the  use  of  the  word  fluxion  and  the  notation  x  by  all  the  English  writers 
previous  to  1704,  excepting  Newton  and  George  Cheyne,  in  the  sense 
of  an  infinitely  small  increment.^  Strange  to  say,  even  in  the  Com- 
mcrcium  epistolicum  the  words  moment  and  fluxion  appear  to  be  used 
as  synonymous. 

After  showing  by  examples  how  to  solve  the  first  problem,  Newton 
proceeds  to  the  demonstration  of  his  solution: — 

"The  moments  of  flowing  quantities  (that  is,  their  indefinitely 
small  parts,  by  the  accession  of  which,  in  infinitely  small  portions  of 
time,  they  are  continually  increased)  are  as  the  velocities  of  their 
flowing  or  increasing. 

"Wherefore,  if  the  moment  of  any  one  (as  x)  be  represented  by  the 
product  of  its  celerity  x  into  an  infinitely  small  quantity  o  (f.  e.  by 

'A.  De  Morgan,  "On  the  Early  History  of  Infinitesimals,"  in  Philosophical 
Magazine.,  November,  1852. 


NEWTON  TO  RULER  tq^ 

xo).  the  moments  of  the  others,  v,  y,  z,  will  be  represented  by  vo,  yo, 
so;  because  vo,  xo,  yo,  and  so  are  to  each  other  as  v,  x,  y,  and  z. 

"Now  since  the  moments,  as  xo  and  yo,  are  the  indefinitely  little 
accessions  of  the  flowing  quantities  x  and  y,  by  which  those  quantities 
are  increased  through  the  several  indefinitely  Httle  intervals  of  time, 
it  follows  that  those  quantities,  x  and  y,  after  any  indefinitely  small 
interval  of  time,  become  x+xo  and  y+yo,  and  therefore  the  equation, 
which  at  all  times  indifferently  expresses  the  relation  of  the  flowing 
quantities,  will  as  well  express  the  relation  between  x+'xo  and  y+yo, 
as  between  x  and  y;  so  that  x+xo  and  y+yo  may  be  substituted  in 
the  same  equation  for  those  quantities,  instead  of  x  and  y.  Thus  let 
any  equation  x^  —  ax~+axy-y^=o  be  given,  and  substitute  x+xo  for 
x,  and  y+yo  for  y,  and  there  will  arise 


—ax-— 2  axxo — axoxo 
+axy+ayxo   +axoyo 

+axyo 
-y^  -3y-yo-3yyoyo-y2o^ 

"Now,  by  supposition,  x^  -ax-+axy  —  y^  =  o,  which  therefore,  being 
expunged  and  the  remaining  terms  being  divided  by  o,  there  will 
remain 

3.T-.T  —  2axx+ayx+axy  —  ^y-y+T,xxxo  —  ax:.xo+axyo  -  T,yyyo 
+x^oo—y^oo=o. 
But  whereas  zero  is  supposed  to  be  infinitely  little,  that  it  may  repre- 
sent the  moments  of  quantities,  the  terms  that  are  multiplied  by  it 
will  be  nothing  in  respect  of  the  rest  (termini  in  earn  ducti  pro  nihilo 
possunt  haberi  cum  aliis  collati);  therefore  I  reject  them,  and  there 
remains 

T^x-x  —  2axx+ayx+axy  —  3y  "y = o, 

as  above  in  Example  I."    Newton  here  uses  infinitesimals. 

Much  greater  than  in  the  first  problem  were  the  difficulties  en- 
countered in  the  solution  of  the  second  problem,  involving,  as  it  does, 
inverse  operations  which  have  been  taxing  the  skill  of  the  best  ana- 
lysts since  his  time.  Newton  gives  first  a  special  solution  to  the  second 
problem  in  which  he  resorts  to  a  rule  for  which  he  has  given  no  proof. 

In  the  general  solution  of  his  second  problem,  Newton  assumed 
homogeneity  with  respect  to  the  fluxions  and  then  considered  three 
cases:  (i)  when  the  equation  contains  two  fluxions  of  quantities  and 
but  one  of  the  fluents;  (2)  when  the  equation  involves  both  the  lluents 
as  well  as  both  the  fluxions;  (3)  when  the  equation  contains  the  flu- 
ents and  the  fluxions  of  three  or  more  quantities.    The  first  case  is  the 

i^asiest  since  it  requires  simply  the  integration  of  -j  =f{x),  to  which 


196  A  HISTORY  OF  MATHEMATICS 

his  "special  solution"  is  applicable.  The  second  case  demanded 
nothing  less  than  the  general  solution  of  a  differential  equation  of  the 
first  order.  Those  who  know  what  efforts  were  afterwards  needed 
for  the  complete  exploration  of  this  field  in  analysis,  will  not  depre- 
ciate Newton's  work  even  though  he  resorted  to  solutions  in  form  of 
infinite  series.  Newton's  third  case  comes  now  under  the  solution  of 
partial  differential  equations.  He  took  the  equation  2x—z^xy=o 
and  succeeded  in  finding  a  particular  integral  of  it. 

The  rest  of  the  treatise  is  devoted  to  the  determination  of  maxima 
and  minima,  the  radius  of  curvature  of  curves,  and  other  geometrical 
applications  of  his  fluxionary  calculus.  All  this  was  done  previous 
to  the  year  1672. 

It  must  be  observed  that  in  the  Method  of  Fluxions  (as  well  as  in 
his  De  Analysi  and  all  earlier  papers)  the  method  employed  by  New- 
ton is  strictly  infinitesimal,  and  in  substance  like  that  of  Leibniz. 
Thus,  the  original  conception  of  the  calculus  in  England,  as  well  as 
on  the  Continent,  was  based  on  infinitesimals.  The  fundamental 
principles  of  the  fluxionary  calculus  were  first  given  to  the  world  in 
the  Principia;  but  its  peculiar  notation  did  not  appear  until  published 
in  the  second  volume  of  Wallis'  Algebra  in  1693.  The  exposition 
given  in  the  Algebra  was  a  contribution  of  Newton;  it  rests  on  in- 
finitesimals. In  the  first  edition  of  the  Principia  (1687)  the  descrip- 
tion of  fluxions  is  likewise  founded  on  infinitesimals,  but  in  the  second 
(17 13)  the  foundation  is  somewhat  altered.  In  Book  II,  Lemma  II, 
of  the  first  edition  we  read:  "Cave  tamen  intellexeris  particular 
finitas.  Momenta  quam  primiim  finitce  sunt  magnitudinis,  desinunt 
esse  momenta.  Finlri  enim  repugnat  aliquatenus  perpetuo  eorum 
incremento  vel  decremento.  Intelligenda  sunt  principia  jamjam  nas- 
centia  finitarum  magnitudinum."  In  the  second  edition  the  two 
sentences  which  we  print  in  italics  are  replaced  by  the  following: 
"Particulae  finitae  non  sunt  momenta  sed  quantitates  ipsge  ex  mo- 
mentis  genitce."  Through  the  difficulty  of  the  phrases  in  both  ex- 
tracts, this  much  distinctly  appears,  that  in  the  first,  moments  are 
infinitely  small  quantities.  What  else  they  are  in  the  second  is  not 
clear.  ^  In  the  Quadrature  of  Curves  of  1704,  the  infinitely  small 
quantity  is  completely  abandoned.  It  has  been  shown  that  in  the 
Method  of  Fluxions  Newton  rejected  terms  involving  the  quantity  o, 
because  they  are  infinitely  smafl  compared  with  other  terms.  This 
reasoning  is  unsatisfactory;  for  as  long  as  o  is  a  quantity,  though 
ever  so  smafl,  this  rejection  cannot  be  made  without  affecting  the 
result.  Newton  seems  to  have  felt  this,  for  in  the  Quadrature  of  Curves 
he  remarked  that  "in  mathematics  the  minutest  errors  are  not  to  be 
neglected"  (errores  quam  minimi  in  rebus  mathematicis  non  sunt 
contemnendi). 

The  early  distinction  between  the  system  of  Newton  and  Leibniz 
1  A.  De  Morgan,  loc.  cit.,  1852. 


NEWTON  TO  EULER 


197 


lies  in  this,  that  Newton,  holding  to  the  conception  of  velocity  or 
fluxion,  used  the  infinitely  small  increment  as  a  means  of  determining 
it,  while  with  Leibniz  the  relation  of  the  infinitely  small  increments 
is  itself  the  object  of  determination.  The  difference  between  the  two 
rests  mainly  upon  a  difference  in  the  mode  of  generating  quantities. 

We  give  Newton's  statement  of  the  method  of  fluxions  or  rates,  as 
given  in  the  introduction  to  his  Quadrature  of  Curves.  "I  consider 
mathematical  quantities  in  this  place  not  as  consisting  of  very  small 
parts,  but  as  described  by  a  continued  motion.  Lines  are  described, 
and  thereby  generated,  not  by  the  apposition  of  parts,  but  by  the 
continued  motion  of  points;  superficies  by  the  motion  of  lines;  solids 
by  the  motion  of  superficies;  angles  by  the  rotation  of  the  sides; 
portions  of  time  by  continual  flux:  and  so  on  in  other  quantities. 
These  geneses  really  take  place  in  the  nature  of  things,  and  are  daily 
seen  in  the  motion  of  bodies.  .  .  . 

"Fluxions  are,  as  near  as  we  please  (quam  proxime),  as  the  incre- 
ments of  fluents  generated  in  times,  equal  and  as  small  as  possible, 
and  to  speak  accurately,  they  are  in  the  prime  ratio  of  nascent  in- 
crements; yet  they  can  be  expressed  by  any  lines  whatever,  which  are 
proportional  to  them." 

Newton  exemplifies  this  last  assertion  by  the  problem  of  tangency: 
Let  AB  he  the  abscissa,  BC  the  ordinate,  VCH  the  tangent,  Ec  the 
increment  of  the  ordinate,  which  produced  meets  VH  at  T,  and  Cc 
the  increment  of  the  curve.  The  right  line  Cc  being  produced  to  K, 
there  are  formed  three  small  triangles,  the  rectilinear  CEc,  the  mix- 
tilinear  CEc,  and  the  rectilinear  CET.  Of  these,  the  first  is  evidently 
the  smallest,  and  the  last  the  greatest.  Now  suppose  the  ordinate  be 
to  move  into  the  place  BC,  so  that  the  point  c  exactly  coincides  with 
the  point  C;  CK,  and 
therefore  the  curve  Cc, 
is  coincident  with  the  tan- 
gent CH,  Ec  is  absolutely 
equal  to  ET,  and  the 
mixtilinear  evanescent  tri- 
angle CEc  is,  in  the  last 
form,  similar  to  the  tri- 
angle CET;  and  its  eva- 
nescent sides  CE,  Ec,  Cc, 
will  be  proportional  to 
CE,  ET,  and  CT,  the 
sides  of  the  triangle  CET. 

Hence  it  follows  that  the  fluxions  of  the  lines  AB,  BC,  AC,  being  in 
the  last  ratio  of  their  evanescent  increments,  are  proportional  to  the 
sides  of  the  triangle  CET,  or,  which  is  all  one,  of  the  triangle  VBC 
similar  theieunto.  As  long  as  the  points  C  and  c  are  distant  from 
each  other  by  an  interval,  however  small,  the  line  CK  will  stand 


198  A  HISTORY  OF  MATHEMATICS 

apart  by  a  small  angle  from  the  tangent  CH.  But  when  CK  co- 
incides with  CH,  and  the  lines  CE,  Ec,  cC  reach  their  ultimate 
ratios,  then  the  points  C  and  c  accurately  coincide  and  are  one 
and  the  same.  Newton  then  adds  that  "in  mathematics  the 
minutest  errors  are  not  to  be  neglected."  This  is  plainly  a  re- 
jection of  the  postulates  of  Leibniz.  The  doctrine  of  infinitely 
small  quantities  is  here  renounced  in  a  manner  which  would  lead 
one  to  suppose  that  Newton  had  never  held  it  himself.  Thus  it 
appears  that  Newton's  doctrine  was  different  in  different  periods. 
Though,  in  the  above  reasoning,  the  Charybdis  of  infinitesimals  is 
safely  avoided,  the  dangers  of  a  Scylla  stare  us  in  the  face.  We  are 
required  to  believe  that  a  point  may  be  considered  a  triangle,  or  that 
a  triangle  can  be  inscribed  in  a  point;  nay,  that  three  dissimilar  tri- 
angles become  similar  and  equal  when  they  have  reached  their  ulti- 
mate form  in  one  and  the  same  point. 

In  the  introduction  to  the  Quadrature  of  Curves  the  fluxion  of  x^ 
is  determined  as  follows: — 

"In  the  same  time  that  x,  by  flowing,  becomes  x-\-o,  the  power 
x'^  becomes  {x+oY,  i.  e.  by  the  method  of  infinite  series 


■n 


x^+nox"   'H ou""   ^-Hetc, 

2 

and  the  increments 

n^ — n  o 
o  and  wox""^-! o-x"   ^H-etc, 

2 

are  to  one  another  as 


I  towx""^-! ox''~^+etc. 

2 


"Let  now  the  increments  vanish,  and  their  last  proportion  will  be 
I  to  «x"~':  hence  the  fluxion  of  the  quantity  x  is  to  the  fluxion  of  the 
quantity  x"  as  i:  »x''~^ 

"The  fluxion  of  hues,  straight  or  curved,  in  all  cases  whatever,  as 
also  the  fluxions  of  superficies,  angles,  and  other  quantities,  can  be 
obtained  in  the  same  manner  by  the  method  of  prime  and  ultimate 
ratios.  But  to  establish  in  this  way  the  analysis  of  infinite  quantities, 
and  to  investigate  prime  and  ultimate  ratios  of  finite  quantities,  nas- 
cent or  evanescent,  is  in  harmony  with  the  geometry  of  the  ancients; 
and  I  have  endeavored  to  show  that,  in  the  method  of  fluxions,  it  is 
not  necessary  to  introduce  into  geometry  infinitely  small  quantities." 
This  mode  of  differentiating  does  not  remove  all  the  difficulties  con- 
nected with  the  subject.    When  o  becomes  nothing.,  then  we  get  the 

ratio  -=nx^~'^,  which  needs  further  elucidation.    Indeed,  the  method 
o 

of  Newton,  as  delivered  by  himself,  is  encumbered  with  difficulties 


NEWTON  TO  EULER  199 

and  objections.  Later  we  shall  state  Bishop  Berkeley's  objection  to 
this  reasoning.  Even  among  the  ablest  admirers  of  Newton,  there 
have  been  obstinate  disputes  respecting  his  explanation  of  his  method 
of  "prime  and  ultimate  ratios." 

The  so-called  "method  of  limits"  is  frequently  attributed  to  New- 
ton, but  the  pure  method  of  limits  was  never  adopted  by  him  as  his 
method  of  constructing  the  calculus.  All  he  did  was  to  estabHsh  in 
his  Principia  certain  principles  which  are  applicable  to  that  method, 
but  which  he  used  for  a  different  purpose.  The  first  lemma  of  the 
first  book  has  been  made  the  foundation  of  the  method  of  hmits: — 

"Quantities  and  the  ratios  of  quantities,  which  in  any  finite  time 
converge  continually  to  equality,  and  before  the  end  of  that  time  ap- 
proach nearer  the  one  to  the  other  than  by  any  given  difference,  be- 
come ultimately  equal." 

In  this,  as  well  as  in  the  lemmas  following  this,  there  are  obscurities 
and  difficulties.  Newton  appears  to  teach  that  a  variable  quantity 
and  its  limit  will  ultimately  coincide  and  be  equal. 

The  full  title  of  Newton's  Principia  is  Philosophice  Natiiralis  Prin- 
cipia Mathematica.  It  was  printed  in  16S7  under  the  direction,  and 
at  the  expense,  of  Edmund  Halley.  A  second  edition  was  brought 
out  in  1 7 13  with  many  alterations  and  improvements,  and  accom- 
panied by  a  preface  from  Roger  Cotes.  It  was  sold  out  in  a  few 
months,  but  a  pirated  edition  published  in  Amsterdam  supplied  the 
demand.  The  third  and  last  edition  which  appeared  in  England  during 
Newton's  lifetime  was  published  in  1726  by  Henry  Pemberton.  The 
Principia  consists  of  three  books,  of  which  the  first  two,  constituting 
the  great  bulk  of  the  work,  treat  of  the  mathematical  principles  of 
natural  philosophy,  namely,  the  laws  and  conditions  of  motions  and 
forces.  In  the  third  book  is  drawn  up  the  constitution  of  the  universe 
as  deduced  from  the  foregoing  principles.  The  great  principle  under- 
lying this  memorable  work  is  that  of  universal  gravitation.  The  first 
book  was  completed  on  April  28,  16S6.  After  the  remarkably  short 
period  of  three  months,  the  second  book  was  finished.  The  third  book 
is  the  result  of  the  next  nine  or  ten  months'  labors.  It  is  only  a  sketch 
of  a  much  more  extended  elaboration  of  the  subject  which  he  had 
planned,  but  which  was  never  brought  to  completion. 

The  law  of  gravitation  is  enunciated  in  the  first  book.  Its  discovery 
envelops  the  name  of  Newton  in  a  halo  of  perpetual  glory.  The  cur- 
rent version  of  the  discovery  is  as  follows:  it  was  conjectured  by 
Robert  Hooke  (1635-1703),  C.  Huygens,  E.  Halley,  C.  Wren,  I.  New- 
ton, and  others,  that,  if  J.  Kepler's  third  law  was  true  (its  absolute 
accuracy  was  doubted  at  that  time),  then  the  attraction  between  the 
earth  and  other  members  of  the  solar  system  varied  inversely  as  the 
square  of  the  distance.  But  the  proof  of  the  truth  or  falsity  of  the 
guess  was  wanting.  In  1666  Newton  reasoned,  in  substance,  that  if 
g  represent  the  acceleration  of  gravity  on  the  surface  of  the  earth,  ri 


230  A  HISTORY  OF  MATHEMATICS 

be  the  earth's  radius,  R  the  distance  of  the  moon  from  the  earth,  T 
the  time  of  lunar  revolution,  and  a  a  degree  at  the  equator,  then,  if 
the  law  is  true, 


g^2=47r  y.'  or  g=  jr,(^-j  •  i8oa. 


The  data  at  Newton's  command  gave  i?  =  6o.4r,  r=  2,360,628  seconds, 
but  a  only  60  instead  of  6gj  English  miles.  This  wrong  value  of  a 
rendered  the  calculated  value  of  g  smaller  than  its  true  value,  as 
known  from  actual  measurement.  It  looked  as  though  the  law  of 
inverse  squares  were  not  the  true  law,  and  Newton  laid  the  calculation, 
aside.  In  1684  he  casually  ascertained  at  a  meeting  of  the  Royal 
Society  that  Jean  Picard  had  measured  an  arc  of  the  meridian,  and 
obtained  a  more  accurate  value  for  the  earth's  radius.  Taking  the 
corrected  value  for  a,  he  found  a  figure  for  g  which  corresponded  to 
the  known  value.  Thus  the  lav/  of  inverse  squares  was  verified.  In  a 
scholium  in  the  Principia,  Newton  acknowledged  his  indebtedness  to 
Huygens  for  the  laws  on  centrifugal  force  employed  in  his  calculation. 

The  perusal  by  the  astronomer  Adams  of  a  great  mass  of  unpub- 
lished letters  and  manuscripts  of  Newton  forming  the  Portsmouth 
collection  (which  remained  private  property  until  1872,  when  its 
owner  placed  it  in  the  hands  of  the  University  of  Cambridge)  seems  to 
indicate  that  the  difficulties  encountered  by  Newton  in  the  above 
calculation  were  of  a  dififerent  nature.  According  to  Adams,  Newton's 
numerical  verification  was  fairly  complete  in  1666,  but  Newton  had 
not  been  able  to  determine  what  the  attraction  of  a  spherical  shell 
upon  an  external  point  would  be.  His  letters  to  E.  Halley  show 
thac  he  did  not  suppose  the  earth  to  attract  as  though  all  its  mass 
were  concentrated  into  a  point  at  the  centre.  He  could  not  have 
asserted,  therefore,  that  the  assumed  law  of  gravity  was  verified  by 
the  figures,  though  for  long  distances  he  might  have  claimed  that  it 
yielded  close  approximations.  When  Halley  visited  Newton  in  1684, 
he  requested  Newton  to  determine  what  the  orbit  of  a  planet  would 
be  if  the  law  of  attraction  were  that  of  inverse  squares.  Newton  had 
solved  a  similar  problem  for  R.  Hooke  in  1679,  and  replied  at  once 
that  it  was  an  ellipse.  After  Halley's  visit,  Newton,  with  Picard's 
new  value  for  the  earth's  radius,  reviewed  his  early  calculation,  and 
was  able  to  show  that  if  the  distances  between  the  bodies  in  the  solar 
system  were  so  great  that  the  bodies  might  be  considered  as  points, 
then  their  motions  were  in  accordance  with  the  assumed  law  of  gravi- 
tation. In  1685  he  completed  his  discovery  by  showing  that  a  sphere 
whose  density  at  any  point  depends  only  on  the  distance  from  the 
centre  attracts  an  external  point  as  though  its  whole  mass  were  con- 
centrated at  the  centre. 

Newton's  unpublished  manuscripts  in  the  Portsmouth  collection 
show  that  he  had  worked  out,  by  means  of  fluxions  and  fluents,  his 


NEVV'ION    10  EULER  201 

lunar  calculations  to  a  higher  degree  of  approximation  than  ihat  given 
in  the  Principia,  but  that  he  was  unable  to  interpret  his  results  geo- 
metrically. The  papers  in  that  collection  throw  light  upon  the  mode 
by- which  Newton  arrived  at  some  of  the  results  in  the  Principia,  as, 
for  instance,  the  famous  solution  in  Book  II,  Prop.  35,  SchoUum,  of 
the  problem  of  the  solid  of  revolution  which  moves  through  a  resisting 
medium  with  the  least  resistance.  The  solution  is  unproved  in  the 
Principia,  but  is  demonstrated  by  Newton  in  the  draft  of  a  letter  to 
David  Gregory  of  Oxford,  found  in  the  Portsmouth  collection.^ 

It  is  chiefly  upon  the  Principia  that  the  fame  of  Newton  rests. 
David  Brewster  calls  it  "the  brightest  page  in  the  records  of  human 
reason."  Let  us  listen,  for  a  moment,  to  the  comments  of  P.  S.  La- 
place, the  foremost  among  those  followers  of  Newton  who  grappled 
with  the  subtle  problems  of  the  motions  of  planets  under  the  influence 
of  gravitation:  "Newton  has  well  established  the  existence  of  the 
principle  which  he  had  the  merit  of  discovering,  but  the  development 
of  its  consequences  and  advantages  has  been  the  work  of  the  successors 
of  this  great  mathematician.  The  imperfection  of  the  infinitesimal 
calculus,  when  first  discovered,  did  not  allow  him  completely  to  re- 
solve the  difficult  problems  which  the  theory  of  the  universe  offers; 
and  he  was  oftentimes  forced  to  give  mere  hints,  which  were  always 
uncertain  till  confirmed  by  rigorous  analysis.  Notwithstanding  these 
unavoidable  defects,  the  importance  and  the  generality  of  his  dis- 
coveries respecting  the  system  of  the  universe,  and  the  most  interesting 
points  of  natural  philosophy,  the  great  number  of  profound  and  orig- 
inal views,  which  have  been  the  origin  of  the  most  brilliant  discoveries 
of  the  mathematicians  of  the  last  century,  which  were  all  presented 
with  much  elegance,  will  insure  to  the  Principia  a  lasting  pre-eminence 
over  all  other  productions  of  the  human  mind." 

Newton's  Arithmetica  universalis,  consisting  of  algebraical  lectures 
delivered  by  him  during  the  first  nine  years  he  was  professor*  at  Cam- 
bridge, were  published  in  1707,  or  more  than  thirty  years  after  they 
were  written.  This  work  was  published  by  William  Whiston  (1667- 
1752).  We  are  not  accurately  informed  how  Whiston  came  in  pos- 
session of  it,  but  according  to  some  authorities  its  publication  was  a 
breach  of  confidence  on  his  part.  He  succeeded  Newton  in  the 
Lucasian  professorship  at  Cambridge. 

The  Arithmetica  universalis  contains  new  and  important  results  on 
the  theory  of  equations.  Newton  states  Descartes'  rule  of  signs  in 
accurate  form  and  gives  formulae  expressing  the  sum  of  the  powers 
of  the  roots  up  to  the  sixth  power  and  by  an  "and  so  on"  makes  it 
evident  that  they  can  be  extended  to  any  higher  power.  Newton's 
formulae  take  the  implicit  form,  while  similar  formulae  given  earlier 

'O.  Bolza,  in  Bihliolheca  mathemalica,  3.  S.,  Vol.  13,  1913,  pp.  146-149.  For 
a  bibliography  of  this  "problem  of  Newton"  on  the  surface  of  least  resistance,  see 
L'InicrmeJiairc  dis  inMhematicicns,  Vol.  23,  1916,  pp.  81-S4. 


202  A  HISTORY  OF  MATHEMATICS 

by  Albert  Girard  take  the  explicit  form,  as  do  also  the  general  formulas 
derived  later  by  E.  Waring.  Newton  uses  his  formulas  for  fixing  an 
upper  limit  of  real  roots;  the  sum  of  any  even  power  of  all  the  roots 
must  exceed  the  same  even  power  of  any  one  of  the  roots.  He  estab- 
lished also  another  limit:  A  number  is  an  upper  limit,  if,  when  sub- 
stituted for  X,  it  gives  to  f{x)  and  to  all  its  derivatives  the  same  sign. 
In  1748  Colin  Maclaurin  proved  that  an  upper  hmit  is  obtained  by 
adding  unity  to  the  absolute  value  of  the  largest  negative  coefficient 
of  the  equation.  Newton  showed  that  in  equations  with  real  co- 
efficients, imaginary  roots  always  occur  in  pairs.  His  inventive  genius 
is  grandly  displayed  in  his  rule  for  determining  the  inferior  limit  of  the 
number  of  imaginary  roots,  and  the  superior  limits  for  the  number 
of  positive  and  negative  roots.  Though  less  expeditious  than  Des- 
cartes', Newton's  rule  always  gives  as  close,  and  generally  closer, 
limits  to  the  number  of  positive  and  negative  roots,  Newton  did 
not  prove  his  rule. 

Some  light  was  thrown  upon  it  by  George  Campbell  and  Colin 
Maclaurin,  in  the  Philosophical  Transactions,  of  the  years  1728  and 
1729.  But  no  complete  demonstration  was  found  for  a  century  and  a 
half,  until,  at  last,  Sylvester  established  a  remarkable  general  theorem 
which  includes  Newton's  rule  as  a  special  case.  Not  without  interest 
is  Newton's  suggestion  that  the  conchoid  be  admitted  as  a  curve  to 
be  used  in  geometric  constructions,  along  with  the  straight  line  and 
circle,  since  the  conchoid  can  be  used  for  the  duplication  of  a  cube  and 
trisection  of  an  angle — to  one  or  the  other  of  which  every  problem 
involving  curves  of  the  third  or  fourth  degree  can  be  reduced. 

The  treatise  on  Method  of  Fluxions  contains  Newton's  method  of 
approximating  to  the  roots  of  numerical  equations.  Substantially 
the  same  explanation  is  given  in  his  De  analyst  per  aquationes  numero 
terminorum  infinitas.  He  explains  it  by  working  one  example,  namely 
the  now  famous  cubic  ^  y^- 2^-5  =  0.  The  earliest  printed  account 
appeared  in  Wallis'  Algebra,  1685,  chapter  94.  Newton  assumes  that 
an  approximate  value  is  already  known,  which  differs  from  the  true 
value  by  less  than  one-tenth  of  that  value.  He  takes  y  =  2  and  sub- 
stitutes y=2+/)  in  the  equation,  which  becomes  />^+6/>^-|-io/>- 1  =  0. 
Neglecting  the  higher  powers  of  p,  he  gets  10/)— 1  =  0.  Taking 
P=.i+q,  he  gets  q^+6.^q^+ii.2^q+.o6i=o.  From  ii.235'+.o6i  =  o 
he  gets  g=  — .0054+r,  and  by  the  same  process,  r= —.00004853. 
Finally  y  =  2+.i- .0054 -.00004853  =  2.09455147.  Newton  arranges 
his  work  in  a  paradigm.  He  seems  quite  aware  that  his  method  may 
fail.  If  there  is  doubt,  he  says,  whether  p  =  .i  is  sufficiently  close  to 
the  truth,  find  p  from  6p^+iop—i=o.  He  does  not  show  that  even 
this  latter  method  will  always  answer.     By  the  same  mode  of  pro- 

1  For  quotations  from  Newton,  see  F.  Cajori,  "Historical  Note  on  the  Newton- 
Raphson  Method  of  Approximation,"  Amer.  Math.  Monthly,  Vol.  18,  191 1,  pp.  29- 
33- 


NEWTON  TO  EULER  20;, 

cedure,  Newton  finds,  by  a  rapidly  converging  series,  the  value  of  y 
in  terms  of  a  and  .v,  in  the  equation  >'^+o.vy+flay  —  .v^  — 20^  =  0. 

In  1690,  Joseph  Raphson  (164S-1715),  a  fellow  of  the  Royal  Society 
of  London,  published  a  tract,  Analysis  ceqiiationum  universalis.  His 
method  closely  resembles  that  of  Newton.  The  only  difference  is 
this,  that  Newton  derives  each  successive  step,  p,  q,  r,  of  approach  to 
the  root,  from  a  neio  equation,  while  Raphson  fmds  it  each  time  by 
substitution  in  the  original  equation.  In  Newton's  cubic,  Raphson 
would  not  find  the  second  correction  by  the  use  of  a"^4-6.r-+  lo.v  -1=0, 
but  would  substitute  2.1+9  i'^  the  original  equation,  finding  q  = 
-  .0054.  He  would  then  substitute  2.0946+r  in  the  original  equation, 
finding  r= —  .00004853,  and  so  on.  Raphson  does  not  mention 
Newton;  he  evidently  considered  the  difference  sufficient  for  his 
method  to  be  classed  independently.  To  be  emphasized  is  the  fact 
that  the  process  which  in  modern  texts  goes  by  the  name  of  "New- 
ton's method  of  approximation,"  is  really  not  Newton's  method,  but 

Raphson's  modification  of  it.    The  form  now  so  familiar,  a  —  ~-r-  was 

not  used  by  Newton,  but  was  used  by  Raphson.  To  be  sure,  Raphson 
does  not  use  this  notation;  he  writes /(a)  and /'(a)  out  in  full  as  poly- 
nomials. It  is  doubtful,  whether  this  method  should  be  named  after 
Newton  alone.  Though  not  identical  with  Vieta's  process,  it  re- 
sembles Vieta's.  The  chief  difference  lies  in  the  divisor  used.  The 
divisor  f'{a)  is  much  simpler,  and  easier  to  compute  than  Vieta's 
divisor.  Raphson's  version  of  the  process  represents  what  J.  Lagrange 
recognized  as  an  advance  on  the  scheme  of  Newton.  The  method  is 
"plus  simple  que  celle  de  Newton."  ^  Perhaps  the  name  "Newton- 
Raphson  method"  would  be  a  designation  more  nearly  representing 
the  facts  of  history.  We  may  add  that  the  solution  of  numerical 
equations  was  considered  geometrically  by  Thomas  Baker  in  1684 
and  Edmund  Halley  in  1687,  but  in  1694  Halley  "had  a  very  great 
desire  of  doing  the  same  in  numbers."  The  only  difference  between 
Halley's  and  Newton's  own  method  is  that  Halley  solves  a  quadratic 
equation  at  each  step,  Newton  a  linear  equation.  Halley  modified 
also  certain  algebraic  expressions  yielding  approximate  cube  and 
fifth  roots,  given  in  1692  by  the  Frenchman,  Thomas  Fantct  dc  Lagny 
(1660-1734).  In  1705  and  1706  Lagny  outlines  a  method  of  differences 
such  a  method,  less  systematically  developed,  had  been  previously 
explained  in  England  by  John  Collins.  By  this  method,  if  a,  ft,  c,  . 
are  in  arithmetical  progression,  then  a  root  may  be  found  approxi- 
mately from  the  first,  second,  and  higher  differences  of  /(c),  f{b) 
fie),  •  .  . 

Newton's  Method  of  Fluxions  contains  also  "Newton's  parallelo- 
gram," which  enabled  him,  in  an  equation, /(.r,  3')  =  o,  to  find  a  series 

1  Lagrange,  Resolution  dcs  eqiiat.  num.,  1798,  Note  V,  p.  13S. 


204  A  HISTORY  OF  MATHEMATICS 

in  powers  of  x  equal  to  the  variable  y.  The  great  utility  of  this  rule 
lay  in  its  determining  the  form  of  the  series;  for,  as  soon  as  the  law  was 
known  by  which  the  exponents  in  the  series  vary,  then  the  expansion 
could  be  effected  by  the  method  of  indeterminate  coefficients.  The 
rule  is  still  used  in  determining  the  infinite  branches  to  curves,  or  their 
figure  at  multiple  points.  Newton  gave  no  proof  for  it,  nor  any  clue 
as  to  how  he  discovered  it.  The  proof  was  supplied  half  a  century 
later,  by  A.  G.  Kastner  and  G.  Cramer,  independently.^ 

In  1704  was  published,  as  an  appendix  to  the  Opticks,  the  Enu- 
meratio  lincanim  tertii  ordinis,  which  contains  theorems  on  the  theory 
of  curves.  Newton  divides  cubics  into  seventy-two  species,  arranged 
in  larger  groups,  for  which  his  commentators  have  supplied  the  names 
"genera"  and  "classes,"  recognizing  fourteen  of  the  former  and  seven 
(or  four)  of  the  latter.  He  overlooked  six  species  demanded  by  his 
principles  of  classification,  and  afterwards  added  by  J.  Stirling,  Wil- 
liam Murdoch  (1754-1839),  and  G.  Cramer.  He  enunciates  the  re- 
markable theorem  that  the  five  species  which  he  names  "divergent 
parabolas"  give  by  their  projection  every  cubic  curve  whatever.  As 
a  rule,  the  tract  contains  no  proofs.  It  has  been  the  subject  of  frequent 
conjecture  how  Newton  deduced  his  results.  Recently  we  have  gotten 
at  the  facts,  since  much  of  the  analysis  used  by  Newton  and  a  few 
additional  theorems  have  been  discovered  among  the  Portsmouth 
papers.  An  account  of  the  four  holograph  manuscripts  on  this  sub- 
ject has  been  published  by  W.  W.  Rouse  Ball,  in  the  Transactions  of 
the  London  Mathe?natical  Society  (vol.  xx,  pp.  104-143).  It  is  inter- 
esting to  observe  how  Newton  begins  his  research  on  the  classification 
of  cubic  curves  by  the  algebraic  method,  but,  finding  it  laborious, 
attacks  the  problem  geometrically,  and  afterwards  returns  again  to 
analysis. 

Space  does  not  permit  us  to  do  more  than  merely  mention  Newton's 
prolonged  researches  in  other  departments  of  science.  He  conducted 
a  long  series  of  experiments  in  optics  and  is  the  author  of  the  corpus- 
cular theory  of  light.  The  last  of  a  number  of  papers  on  optics, 
which  he  contributed  to  the  Royal  Society,  1687,  elaborates  the  theory 
of  "fits."  He  explained  the  decomposition  of  light  and  the  theory 
of  the  rainbow.  By  him  were  invented  the  reflecting  telescope  and 
the  sextant  (afterwards  re-invented  by  Thomas  Godfrey  of  Phila- 
delphia ^  and  by  John  Hadley).  He  deduced  a  theoretical  expression 
for  the  velocity  of  sound  in  air,  engaged  in  experiments  on  chemistry, 
elasticity,  magnetism,  and  the  law  of  cooling,  and  entered  upon  geo- 
logical speculations. 

During  the  two  years  following  the  close  of  1692,  Newton  suffered 

1  S.  Giinther,  Vermischte  U ntcrsuchiingen  zur  Geschichte  d.  math.  Wiss.,  Leipzig' 
1876,  pp.  136-187. 

2F.  Cajori,  Teaching  and  History  of  Mathematics  in  the  U.  S.,  Washington,  1890, 
p.  42. 


NEWrON  TO  EULER  205 

from  insomnia  and  nervous  irritability.  Some  thought  that  he  la- 
bored under  temporary  mental  aberration.  Though  he  recovered  his 
tranquillity  and  strength  of  mind,  the  time  of  great  discoveries  was 
over;  he  would  study  out  questions  propounded  to  him,  but  no  longer 
did  he  by  his  own  accord  enter  upon  new  fields  of  research.  The 
most  noted  investigation  after  his  sickness  was  the  testing  of  his  lunar 
theory  by  the  observations  of  Flamsteed,  the  astronomer  royal.  In 
1695  he  was  appointed  warden,  and  in  1690  master  of  the  mint,  which 
office  he  held  until  his  death.  His  body  was  interred  in  Westminster 
Abbey,  where  in  1731  a  magnificent  monument  was  erected,  bearing 
an  inscription  ending  with,  "Sibi  gratulentur  mortales  tale  tantumque 
exstitisse  humani  generis  decus."  It  is  not  true  that  the  Binomial 
Theorem  is  also  engraved  on  it. 

We  pass  to  Leibniz,  the  second  and  independent  inventor  of  the 
calculus.  Gottfried  Wilhelm  Leibniz  (1646-17 16)  was  born  in  Leip- 
zig. No  period  in  the  history  of  any  civilized  nation  could  have  been 
less  favorable  for  literary  and  scientific  pursuits  than  the  middle  of 
the  seventeenth  century  in  Germany.  Yet  circumstances  seem  to 
have  happily  combined  to  bestow  on  the  youthful  genius  an  education 
hardly  otherwise  obtainable  during  this  darkest  period  of  German 
history.  He  was  brought  early  in  contact  with  the  best  of  the  culture 
then  existing.  In  his  fifteenth  year  he  entered  the  University  of 
Leipzig.  Though  law  was  his  principal  study,  he  applied  himself 
with  great  diligence  to  every  branch  of  knowledge.  Instruction  in 
German  universities  was  then  very  low.  The  higher  mathematics 
was  not  taught  at  all.  We  are  told  that  a  certain  John  Kuhn  lectured 
on  Euclid's  Elements,  but  that  his  lectures  were  so  obscure  that  none 
except  Leibniz  could  understand  them.  Later  on,  Leibniz  attended, 
for  a  half-year,  at  Jena,  the  lectures  of  Erhard  Weigel,  a  philosopher 
and  mathematician  of  local  reputation.  In  1666  Leibniz  published 
a  treatise,  De  Arte  Contbinatoria,  in  which  he  does  not  pass  beyond 
the  rudiments  of  mathematics,  but  which  contains  remarkable  plans 
for  a  theory  of  mathematical  logic,  a  s>mibolic  method  with  formal 
rules  obviating  the  necessity  of  thinking.  Vaguely  such  plans  had 
been  previously  suggested  by  R.  Descartes  and  Pierre  Herigone.  In 
manuscripts  which  Leibniz  left  unpublished  he  enunciated  the  princi- 
pal properties  of  what  is  now  called  logical  multiplication,  addition, 
negation,  identity,  class-induction  and  the  null-class.^  Other  theses 
written  by  him  at  this  time  were  metaphysical  and  juristical  in  char- 
acter. A  fortunate  circumstance  led  Leibniz  abroad.  In  1672  he  was 
sent  by  Baron  Boineburg  on  a  political  mission  to  Paris.  He  there 
formed  the  acquaintance  of  the  most  distinguished  men  of  the  age. 
Among  these  was  C.  Huygens,  who  presented  a  copy  of  his  work  on 
the  oscillation  of  the  pendulum  to  Leibniz,  and  first  led  the  gifted 
young  German  to  the  study  of  higher  mathematics.  In  1673  Leibniz 
*  See  Philip  E.  B.  Jourdain  in  Quarterly  Jour,  of  Math.,  Vol.  41,  1910,  p.  329. 


2o6  A  HISTORY  OF  MATHEMATICS 

went  to  London,  and  remained  there  from  January  till  March.  He 
there  became  incidentally  acquainted  with  the  mathematician  John 
Pell,  to  whom  he  explained  a  method  he  had  found  on  the  summation 
of  series  of  numbers  by  their  differences.  Pell  told  him  that  a  similar 
formula  had  been  published  by  Gabriel  Mouton  (1618-1694)  as 
early  as  1670,  and  then  called  his  attention  to  N.  Mercator's  work 
on  the  rectification  of  the  parabola.  While  in  London,  Leibniz  ex- 
hibited to  the  Royal  Society  his  arithmetical  machine,  which  was 
similar  to  B.  Pascal's,  but  more  eflScient  and  perfect.  After  his  re- 
turn to  Paris,  he  had  the  leisure  to  study  mathematics  more  system- 
atically. With  indomitable  energy  he  set  about  removing  his  igno- 
rance of  higher  mathematics.  C.  Huygens  was  his  principal  master. 
He  studied  the  geometric  works  of  R.  Descartes,  Honorarius  Fabri, 
Gregory  St.  Vincent,  and  B.  Pascal.  A  careful  study  of  infinite 
series  led  him  to  the  discovery  of  the  following  expression  for  the 
ratio  of  the  circumference  to  the  diameter  of  the  circle,  previously 
discovered  by  James  Gregory: — 

-  =  i-¥+^-T+¥-etc. 

4 
This  elegant  series  was  found  in  the  same  way  as  N.  Mercator's  on 
the  hyperbola.     C.  Huygens  was  highly  pleased  with  it  and  urged 
him  on  to  new  investigations.    In  1673  Leibniz  derived  the  series 

from  which  most  of  the  practical  methods  of  computing  t  have  been 
obtained.  This  series  had  been  previously  discovered  by  James 
Gregory,  and  was  used  by  Abraham  Sharp  (1651-1742)  under  in- 
structions from  E.  Halley  for  calculating  tt  to  72  places.  In  1706 
John  Machin  (1680-1751),  professor  of  astronomy  at  Gresham  Col- 
lege in  London,  obtained  100  places  by  using  an  expression  that  is 
obtained  from  the  relation 

-  =4  arc  tan  ^— arc  tan  2T9, 
4 
by  substituting  Gregory's  infinite  series  for 

arc  tan  ^  and  arc  tan  2-^9 . 
Machin's  formula  was  used  in  1874  by  William  Shanks  (181 2-1882) 
for  computing  7r  to  707  places. 

Leibniz  entered  into  a  detailed  study  of  the  quadrature  of  curves 
and  thereby  became  intimately  acquainted  with  the  higher  math- 
ematics. Among  the  papers  of  Leibniz  is  still  found  a  manuscript 
on  quadratures,  written  before  he  left  Paris  in  1676,  but  which  was 
never  printed  by  him.  The  more  important  parts  of  it  were  embodied 
in  articles  published  later  in  the  Acta  eruditorum. 

In  the  study  of  Cartesian  geometry  the  attention  of  Leibniz  was 


NEWTON  TO  EULER  207 

drawn  early  to  the  direct  and  inverse  problems  of  tangents.  The 
direct  problem  had  been  solved  by  Descartes  for  the  simplest  curves 
only;  while  the  inverse  had  completely  transcended  the  power  of  his 
analysis.  Leibniz  investigated  both  problems  for  any  curve;  he 
constructed  what  he  called  the  triangidum  characteristicum — an 
infinitely  small  triangle  between  the  infinitely  small  part  of  the  curve 
coinciding  with  the  tangent,  and  the  differences  of  the  ordinates  and 
abscissas.  A  curve  is  here  considered  to  be  a  polygon.  The  triangulum 
characteristicum  is  similar  to  the  triangle  formed  by  the  tangent,  the 
ordinate  of  the  point  of  contact,  and  the  sub-tangent,  as  well  as  to 
that  between  the  ordinate,  normal,  and  sub-normal.  It  was  employed 
by  I.  Barrow  in  England,  but  Leibniz  states  that  he  obtained  it  from 
Pascal.  From  it  Leibniz  observed  the  connection  existing  between  the 
direct  and  inverse  problems  of  tangents.  He  saw  also  that  the  latter 
could  be  carried  back  to  the  quadrature  of  curves.  All  these  results 
are  contained  in  a  manuscript  of  Leibniz,  written  in  1673.  One  mode 
used  by  him  in  effecting  quadratures  was  as  follows:  The  rectangle 
formed'  by  a  sub-normal  p  and  an  element  a  {i.  e.  infinitely  small  part 
of  the  abscissa)  is  equal  to  the  rectangle  formed  by  the  ordinate  y 
and  the  element  /  of  that  ordinate;  or  in  symbols,  pa=yl.  But  the 
summation  of  these  rectangles  from  zero  on  gives  a  right  triangle 
equal  to  half  the  square  of  the  ordinate.  Thus,  using  Cavalieri's  no- 
tation, he  gets 

omn.  pa  =  omn.  yl=^  (omn.  meaning  omnia,  all). 

But  y=omn.  I;  hence 


,1    omn.  I' 

omn.  omn.  /-= 

a        2a 


This  equation  is  especially  interesting,  since  it  is  here  that  Leibniz 
first  introduces  a  new  notation.    He  says:  "It  will  be  useful  to  write 

Tfor  omn.,  as    /  /  for  omn.  I,  that  is,  the  sum  of  the  Ts";  he  then 
writes  the  equation  thus: — ■_ 


&fM- 


From  this  he  deduced  the  simplest  integrals,  such  as 

Since  the  symbol  of  summation    /    raises  the  dimensions,  he  con- 
cluded that  the  opposite  calculus,  or  that  of  ditTerences  d,  would 


2oS  A  HISTORY  OF  MATHEMATICS 

lower  them.    Thus,  if    /    l=ya,  then  1  =  ^-     The  symbol  d  was  at 

first  placed  by  Leibniz  in  the  denominator,  because  the  lowering  of 
the  power  of  a  term  was  brought  about  in  ordinary  calculation  by 
division.  The  manuscript  giving  the  above  is  dated  October  2gth. 
1675.^  This,  then,  was  the  memorable  day  on  which  the  notation 
of  the  new  calculus  came  to  be, — a  notation  which  contributed  enor- 
mously to  the  rapid  growth  and  perfect  development  of  the  calculus. 
Leibniz  proceeded  to  apply  his  new  calculus  to  the  solution  of 
certain  problems  then  grouped  together  under  the  name  of  the  In- 
verse Problems  of  Tangents.  He  found  the  cubical  parabola  to  be 
the  solution  to  the  following:  To  find  the  curve  in  which  the  sub- 
normal is  reciprocally  proportional  to  the  ordinate.  The  correctness 
of  his  solution  was  tested  by  him  by  applying  to  the  result  the  method 
of  tangents  of  Baron  Rene  Francois  de  Sluse  (1622-1685)  and  reason- 
ing backwards  to  the  original  supposition.     In  the  solution  of  the 

third  problem  he  changes  his  notation  from  -5  to  the  now  usual  nota- 
tion dx.  It  is  worthy  of  remark  that  in  these  investigations,  Leibniz 
nowhere  explains  the  significance  of  dx  and  dy,  except  at  one  place 

X    . 
in  a  marginal  note:  "Idem  est  dx  et  -,  id  est,  differentia  inter  duas 

X  proximas."  Nor  does  he  use  the  term  differential,  but  always  differ- 
ence. Not  till  ten  years  later,  in  the  Acta  eruditorum,  did  he  give 
further  explanations  of  these  symbols.  What  he  aimed  at  principally 
was  to  determine  the  change  an  expression  undergoes  when  the  sym- 
bol   I    or  (/  is  placed  before  it.    It  may  be  a  consolation  to  students 

wrestling  with  the  elements  of  the  differential  calculus  to  know  that 
it  required  Leibniz  considerable  thought  and  attention  ^  to  determine 

whether  dx  dy  is  the  same  as  d(xy),  and  -7"  the  same  as  d-.     After 
'  ^"  dy  y^ 

considering  these  questions  at  the  close  of  one  of  his  manuscripts,  he 

concluded  that  the  expressions  were  not  the  same,  though  he  could 

not  give  the  true  value  for  each.    Ten  days  later,  in  a  manuscript 

dated  November  21,  1675,  he  found  the  equation  ydx  =dxy—xdy, 

giving  an  expression  for  d{xy),  which  he  observed  to  be  true  for  all 

curves.     He  succeeded  also  in  eliminating  dx  from  a  differential 

equation,  so  that  it  contained  only  dy,  and  thereby  led  to  the  solution 

of  the  problem  under  consideration.     "Behold,  a  most  elegant  way 

1  C.  J.  Gerhardt,  Entdcckung  der  hoheren  Analysis.    Halle,  1855,  p.  125. 

2  C.  J.  Gerhardt,  Entdeckung  der  Dijfcrenzialrechnimg  durch  Leibniz,  Halle,  1848, 
pp.  25,  41. 


NEWTON  TO  EULER  209 

by  which  the  problems  of  the  inverse  method  of  tangents  are  solved, 
or  at  least  are  reduced  to  quadratures!"  Thus  he  saw  clearly  that 
the  inverse  problems  of  tangents  could  be  solved  by  quadratures,  or, 
in  other  words,  by  the  integral  calculus.  In  course  of  a  half-year  he 
discovered  that  the  direct  problem  of  tangents,  too,  yielded  to  the 
power  of  his  new  calculus,  and  that  thereby  a  more  general  solution 
than  that  of  R.  Descartes  could  be  obtained.  He  succeeded  in  solving 
all  the  special  problems  of  this  kind,  which  had  been  left  unsolved 
by  Descartes.  Of  these  we  mention  only  the  celebrated  problem 
proposed  to  Descartes  by  F.  de  Beaune,  viz.  to  find  the  curve  whose 
ordinate  is  to  its  sub-tangent  as  a  given  line  is  to  that  part  of  the 
ordinate  which  lies  between  the  curve  and  a  line  drawn  from  the 
vertex  of  the  cur\-e  at  a  given  inclination  to  the  axis. 

Such  was,  in  brief,  the  progress  in  the  evolution  of  the  new  calculus 
made  by  Leibniz  during  his  stay  in  Paris.  Before  his  departure,  in 
October,  1676,  he  found  himself  in  possession  of  the  most  elementary 
rules  and  formulae  of  the  infinitesimal  calculus. 

From  Paris,  Leibniz  returned  to  Hanover  by  way  of  London  and 
Amsterdam.  In  London  he  met  John  Collins,  who  showed  him  a 
part  of  his  scientific  correspondence.  Of  this  we  shall  speak  later. 
In  Amsterdam  he  discussed  mathematics  with  R.  F.  de  Sluse,  and 
became  satisfied  that  his  own  method  of  constructing  tangents  not 
only  accomplished  all  that  Sluse's  did,  but  even  more,  since  it  could 
be  extended  to  three  variables,  by  which  tangent  planes  to  surfaces 
could  be  found;  and  especially,  since  neither  irrationals  nor  fractions 
prevented  the  immediate  application  of  his  method. 

In  a  paper  of  July  11,  1677,  Leibniz  gave  correct  rules  for  the  dif- 
ferentiation of  sums,  products,  quotients,  powers,  and  roots.  He  had 
given  the  differentials  of  a  few  negative  and  fractional  powers,  as 
early  as  November,  1676,  but  had  made  some  mistakes.    For  dy/ x, 

he  had  given  the  erroneous  value  — r^,  and  in  another  place  the  value 

V-v 

—  \yr-\\  for  d^  occurs  in  one  place  the  wrong  value, 2>  while  a  few 

2 
lines  lower  is  given  —  —3,  its  correct  value. 

In  1682  was  founded  in  Berlin  the  Ada  eruditorum,  a  journal 
sometimes  known  by  the  name  of  Leipzig  Acts.  It  was  a  partial  imi- 
tation of  the  French  Journal  des  Savans  (founded  in  1665),  and  the 
literary  and  scientific  review  published  in  Germany.  Leibniz  was  a 
frequent  contributor.  E.  W.  Tschirnhausen,  who  had  studied  mathe- 
matics in  Paris  with  Leibniz,  and  who  was  familiar  with  the  new 
analysis  of  Leibniz,  published  in  the  Acta  eriiditorum  a  paper  on  quad- 
ratures, which  consists  principally  of  subject-matter  communicated 


no  A  HISTORY  OF  MATHEMATICS 

Dy  Leibniz  to  Tschirnhausen  during  a  controversy  which  they  had 
had  on  this  subject.  Fearing  that  Tschirnhausen  might  claim  as  his 
own  and  pubhsh  tlie  notation  and  rules  of  the  differential  calculus, 
Leibniz  decided,  at  last,  to  make  public  the  fruits  of  his  inventions. 
In  1684,  or  nine  years  after  the  new  calculus  first  dawned  upon  the 
mind  of  Leibniz,  and  nineteen  years  after  Newton  first  worked  at 
fluxions,  and  three  years  before  the  publication  of  Newton's  Principia, 
Leibniz  published,  in  the  Acta  eruditorum,  his  first  paper  on  the  differ- 
ential calculus.  He  was  unwilling  to  give  to  the  world  all  his  treasures, 
but  chose  those  parts  of  his  work  which  were  most  abstruse  and  least 
perspicuous.  This  epoch-making  paper  of  only  six  pages  bears  the 
title:  "Nova  methodus  pro  maximis  et  minimis,  itemque  tangentibus, 
quae  nee  fractas  nee  irrationales  quantitates  moratur,  et  singulare 
pro  illis  calcuU  genus."  The  rules  of  calculation  are  briefly  stated 
without  proof,  and  the  meaning  of  dx  and  dy  is  not  made  clear. 
Printer's  errors  increased  the  difficulty  of  comprehending  the  subject. 
It  has  been  inferred  from  this  that  Leibniz  himself  had  no  definite 
and  settled  ideas  on  this  subject.  Are  dy  and  dx  finite  or  infinitesimal 
quantities?  At  first  they  appear,  indeed,  to  have  been  taken  as  finite, 
when  he  says:  "We  now  call  any  fine  selected  at  random  dx,  then 
we  designate  the  line  which  is  to  dx  as  y  is  to  the  sub-tangent,  by  dy, 
which  is  the  difference  of  y.''  Leibniz  then  ascertains,  by  his  calculus, 
in  what  way  a  ray  of  light  passing  through  two  differently  refracting 
media,  can  travel  easiest  from  one  point  to  another;  and  then  closes 
his  article  by  giving  his  solution,  in  a  few  words,  of  F.  de  Beaune's 
problem.  Two  years  later  (16S6)  Leibniz  published  in  the  Acta 
eruditorum  a  paper  containing  the  rudiments  of  the  integral  calculus. 
The  quantities  dx  and  dy  are  there  treated  as  infinitely  small.  He 
showed  that  by  the  use  of  his  notation,  the  properties  of  curves  could 
be  fully  expressed  by  equations.    Thus  the  equation 

C         dx 

y^V2X-x'+J  ^^^^, 

characterizes  the  cycloid.^ 

The  great  invention  of  Leibniz,  now  made  public  by  his  articles  in 
the  Acta  eruditorum,  made  little  impression  upon  the  mass  of  mathe- 
maticians. In  Germany  no  one  comprehended  the  new  calculus 
except  Tschirnhausen,  who  remained  indifferent  to  it.  The  author's 
statements  were  too  short  and  succinct  to  make  the  calculus  generally 
understood.  The  first  to  take  up  the  study  of  it  were  two  foreigners,— 
the  Scotchman  John  Craig,  and  the  Swiss  Jakob  {James)  Bernoidli. 
The  latter  wrote  Leibniz  a  letter  in  16S7,  wishing  to  be  initiated  into 
the  mysteries  of  the  new  analysis.  Leibniz  was  then  travelUng  abroad, 
so  that  this  letter  remained  unanswered  till  1690.    James  Bernoulli 

'  C.  J.  Gerhardt,  Gcschichte  dcr  Malhcmatik  in  Dciilschland,  Munchen,   1877, 


NEWTON  TO  EULER  21 1 

succeeded,  meanwhile,  by  close  application,  in  uncovering  the  secrets 
of  the  differential  calculus  without  assistance.  He  and  his  brother 
John  proved  to  be  mathematicians  of  exceptional  power.  They  applied 
themselves  to  the  new  science  with  a  success  and  to  an  extent  which 
made  Leibniz  declare  that  it  was  as  much  theirs  as  his.  Leibniz 
carried  on  an  extensive  correspondence  with  them,  as  well  as  with  other 
mathematicians.  In  a  letter  to  John  Bernoulli  he  suggests,  among 
other  things,  that  the  integral  calculus  be  improved  by  reducing  in- 
tegrals back  to  certain  fundamental  irreducible  forms.  The  integra- 
tion of  logarithmic  expressions  was  then  studied.  The  writings  of 
Leibniz  contain  many  innovations,  and  anticipations  of  since  prom- 
nent  methods.  Thus  he  made  use  of  variable  parameters,  laid  the 
foundation  of  analysis  in  situ,  introduced  in  a  manuscript  of  1678  the 
notion  of  determinants  (previously  used  by  the  Japanese),  in  his 
effort  to  simplify  the  expression  arising  in  the  elimination  of  the  un- 
known quantities  from  a  set  of  linear  equations.  He  resorted  to  the 
device  of  breaking  up  certain  fractions  into  the  sum  of  other  fractions 
for  the  purpose  of  easier  integration;  he  explicitly  assumed  the  prin- 
ciple of  continuity;  he  gave  the  first  instance  of  a  ''singular  solution," 
and  laid  the  foundation  to  the  theory  of  envelopes  in  two  papers,  one 
of  which  contains  for  the  first  time  the  terms  co-ordinate  and  axes  of 
co-ordinates.  He  wrote  on  osculating  curves,  but  his  paper  contained 
the  error  (pointed  out  by  John  Bernoulli,  but  not  admitted  by  Leibniz) 
that  an  osculating  circle  will  necessarily  cut  a  curve  in  four  consecutive 
points.  Well  known  is  his  theorem  on  the  wth  differential  coefficient 
of  the  product  of  two  functions  of  a  variable.  Of  his  many  papers  on 
mechanics,  some  are  valuable,  while  others  contain  grave  errors. 
Leibniz  introduced  in  1694  the  use  of  the  word  function,  but  not  in 
the  modern  sense.  Later  in  that  year  Jakob  Bernoulli  used  the  word 
in  the  Leibnizian  sense.  In  the  appendix  to  a  letter  to  Leibniz,  dated 
July  5,  1698,  John  BernouUi  uses  the  word  in  a  more  nearly  modern 
sense:  "earum  [applicatarum]  quaecunque  functiones  per  alias  appli- 
catas  PZ  expressas."  In  17 18  John  Bernoulli  arrives  at  the  definition 
of  function  as  a  "quantity  composed  in  any  manner  of  a  variable  and 
any  constants."  (On  appelle  ici  fonction  d'une  grandeur  variable, 
une  quantite  composee  de  quelque  maniere  que  ce  soit  de  cette  gran- 
deur variable  et  de  constantes.)  ' 

Leibniz  made  important  contributions  to  the  notation  of  mathe- 
matics. Not  only  is  our  notation  of  the  differential  and  integral 
calculus  due  to  him,  but  he  used  the  sign  of  equality  in  writing  pro- 
portions, thus  a:b=c:d.  In  Leibnizian  manuscripts  occurs  ~  for 
"similar"  and  ~  for  "equal  and  similar"  or  "congruent."  '    Says 

'  See  M.  Cantor,  op.  cit.,  Vol.  Ill,  2  Ed.,  1901,  pp.  215,  216,  456,  457;  Encyclo- 
tetlie  des  sciences  mat/ieinaliques,  Tome  II,  Vol.  I,  pp.  3-5. 

2  Leibniz,  Wcrke  Ed.  Gerhardt,  3.  Folge,  Bd.  V,  p.  153.  See  also  J.  Tropfke, 
op.  (it-.  Vol.  IT,  1903,  p.  12. 


212  A  HISTORY  OF  MATHEMATICS 

P.  E.  B.  Jourdain,'  "Leibniz  himself  attributed  all  of  his  mathe- 
matical discoveries  to  his  improvements  in  notation." 

Before  tracing  the  further  development  of  the  calculus  we  shall 
sketch  the  history  of  that  long  and  bitter  controversy  between  English 
and  Continental  mathematicians  on  the  invention  of  the  calculus. 
The  question  was,  did  Leibniz  invent  it  independently  of  Newton,  or 
was  he  a  plagiarist? 

We  must  begin  with  the  early  correspondence  between  the  parties 
appearing  in  this  dispute.  Newton  had  begun  using  his  notation  of 
fluxions  in  1665.^  In  1669  I.  Barrow  sent  John  Collins  Newtv^n's 
tract,  De  Analyst  per  equationes,  etc. 

The  first  visit  of  Leibniz  to  London  extended  from  the  nth  of  Jan- 
uary until  March,  1673.  He  was  in  the  habit  of  committing  to  writing 
important  scientific  communications  received  from  others.  In  1890 
C.  J.  Gerhardt  discovered  in  the  royal  library  at  Hanover  a  sheet  of 
manuscript  with  notes  taken  by  Leibniz  during  this  journey.^  The)" 
are  headed  "Observata  Philosophica  in  itinere  Anglicano  sub  initium 
anni  1673."  The  sheet  is  divided  by  horizontal  lines  into  sections. 
The  sections  given  to  Chymica,  Mechanica,  Magnetica,  Botanica, 
Anatomica,  Medica,  Miscellanea,  contain  extensive  memoranda,  while 
those  devoted  to  m.athematics  have  very  few  notes.  Under  Geo- 
metrica  he  says  only  this:  "Tangentes  omnium  figurarum.  Figurarum 
geometricarum  explicatio  per  motum  puncti  in  moto  lati."  We  sus- 
pect from  this  that  Leibniz  had  read  Isaac  Barrow's  lectures.  Newton 
is  referred  to  only  under  Optica.  Evidently  Leibniz  did  not  obtain  a 
knowledge  of  fluxions  during  this  visit  to  London,  nor  is  it  claimed 
that  he  did  by  his  opponents. 

Various  letters  of  I.  Newton,  J.  Collins,  and  others,  up  to  the  be- 
ginning of  1676,  state  that  Newton  invented  a  method  by  which  tan- 
gents could  be  drawn  without  the  necessity  of  freeing  their  equations 
from  irrational  terms.  Leibniz  announced  in  1674  to  H.  Oldenburg, 
then  secretary  of  the  Royal  Society,  that  he  possessed  very  general 
analytical  methods,  by  which  he  had  found  theorems  of  great  im- 
portance on  the  quadrature  of  the  circle  by  means  of  series.  In  answer, 
Oldenburg  stated  Newton  and  James  Gregory  had  also  discovered 
methods  of  quadratures,  which  extended  to  the  circle.  Leibniz  de- 
sired to  have  these  methods  communicated  to  him;  and  Newton,  at 
the  request  of  Oldenburg  and  Collins,  wrote  to  the  former  the  cele- 
brated letters  of  June  13  and  October  24,  1676.  The  first  contained 
the  Binomial  Theorem  and  a  variety  of  other  matters  relating  to 
infinite  series  and  quadratures;  but  nothing  directly  on  the  method  of 

1  P.  E.  B.  Jourdain,  The  Kalurc  of  Mathematics,  London,  p.  71. 

2  J.  Edleston,  Correspondence  of  Sir  Isaac  Newton  and  Professor  Cotes,  London, 
1850,  p.  xxi;  A.  De  Morgan,  "Fluxions"  and  " Commercium  EpistoUcum "  in 
the  Penny  Cyclopcedia. 

^  C.  J.  Gerhardt,  "Leibniz  in  London"  in  Sitzungsberichte  der  K.  Prcussiscfien 
Academic  d.  Wisscnsch.  z:i  Berlin,  Feb.,  iSot. 


NEWTON  TO  EULER  213 

fluxions.  Leibniz  in  reply  speaks  in  the  highest  terms  of  what  Newton 
had  done,  and  requests  further  explanation.  Newton  in  his  second 
letter  just  mentioned  explains  the  way  in  which  he  found  the  Binomial 
Theorem,  and  also  communicates  his  method  of  fluxions  and  fluents 
in  form  of  an  anagram  in  which  all  the  letters  in  the  sentence  com- 
municated were  placed  in  alphabetical  order.  Thus  Newton  says 
that  his  method  of  drawing  tangents  was 

6accd  CB  12,0 Jf  71 2,1  gn  40  4qrr  45  9/ 1 2vx. 

The  sentence  was,  "Data  a?quatione  quotcunque  fluentes  quantitates 
involvente  fluxiones  invenire,  et  vice  versa."  ("Having  any  given 
equation  involving  never  so  many  flowing  quantities,  to  find  the 
fluxions,  and  vice  versa.")  Surely  this  anagram  afforded  no  hint. 
Leibniz  wrote  a  reply  to  John  Collins,  in  which,  without  any  desire 
of  concealment,  he  explained  the  principle,  notation,  and  the  use  of 
the  differential  calculus. 

The  death  of  Oldenburg  brought  this  correspondence  to  a  close. 
Nothing  material  happened  till  1684,  when  Leibniz  pubhshed  his 
first  paper  on  the  differential  calculus  in  the  Ada  enidilorum,  so  that 
while  Newton's  claim  to  the  priority  of  invention  must  be  admitted 
by  all,  it  must  also  be  granted  that  Leibniz  was  the  first  to  give  the 
full  benefit  of  the  calculus  to  the  world.  Thus,  while  Newton's  in- 
vention remained  a  secret,  communicated  only  to  a  few  friends,  the 
calculus  of  Leibniz  was  spreading  over  the  Continent.  No  rivalry  or 
hostility  existed,  as  yet,  between  the  illustrious  scientists.  Newton 
expressed  a  very  favorable  opinion  of  Leibniz's  inventions,  known  to 
him  through  the  above  correspondence  with  Oldenburg,  in  the  follow- 
ing celebrated  scholium  {Principia,  first  edition,  16S7,  Book  II, 
Prop.  7,  scholium): — 

"In  letters  which  went  between  me  and  that  most  excellent  geom- 
eter, G.  G.  Leibniz,  ten  years  ago,  when  I  signified  that  I  was  in  the 
knowledge  of  a  method  of  determining  maxim- i,  and  minima,  of  draw- 
ing tangents,  and  the  like,  and  when  I  concealf  d  it  in  transposed  letters 
involving  this  sentence  (Data  aequatione,  etc.,  above  cited),  that  most 
distinguished  man  wrote  back  that  he  had  also  fallen  upon  a  method 
of  the  same  kind,  and  communicated  his  method,  which  hardly  dif- 
fered from  mine,  except  in  his  forms  of  words  and  symbols." 

As  regards  this  passage,  we  shall  see  that  Newton  was  afterwards 
weak  enough,  as  De  Morgan  says:  "First,  to  deny  the  plain  and  ob- 
vious meaning,  and  secondly,  to  omit  it  entirely  from  the  third  edition 
of  the  Principia.''  On  the  Continent,  great  progress  was  made  in 
the  calculus  by  Leibniz  and  his  coadjutors,  the  brothers  James  and 
John  Bernoulli,  and  Marquis  de  I'Hospital.  In  1695  John  Wallis  in- 
formed Newton  by  letter  that  "he  had  heard  that  his  notions  of 
fluxions  passed  in  Holland  with  great  applause  by  the  name  of  'Leib- 
niz's Calculus  D'fferentialis.' "    Accordingly  Wallis  stated  in  the  prof- 


214  A  HISTORY  OF  MATHEMATICS 

ace  to  a  volume  of  his  works  that  the  calculus  differentialis  was  New- 
ton's method  of  fluxions  which  had  been  communicated  to  Leibniz 
in  the  Oldenburg  letters.  A  review  of  Wallis'  works,  in  the  Ada 
eruditorum  for  1696,  reminded  the  reader  of  Newton's  own  admission 
in  the  scholium  above  cited. 

For  fifteen  years  Leibniz  had  enjoyed  unchallenged  the  honor  of 
being  the  inventor  of  his  calculus.  But  in  1699  Falio  de  Duillier 
(1664-1753),  a  Swiss,  who  had  settled  in  England,  stated  in  a  mathe- 
matical paper,  presented  to  the  Royal  Society,  his  conviction  that 
L  Newton  was  the  first  inventor;  adding  that,  whether  Leibniz,  the 
second  inventor,  had  borrowed  anything  from  the  other,  he  would 
leave  to  the  judgment  of  those  who  had  seen  the  letters  and  manu- 
scripts of  Newton.  This  was  the  first  distinct  insinuation  of  plagiar- 
ism. It  would  seem  that  the  English  mathematicians  had  for  some 
time  been  cherishing  suspicions  unfavorable  to  Leibniz.  A  feeling 
had  doubtless  long  prevailed  that  Leibniz,  during  his  second  visit  to 
London  in  1676,  had  or  might  have  seen  among  the  papers  of  John 
Collins,  Newton's  Analysis  per  cequationes,  etc.,  which  contained  ap- 
plications of  the  fluxionary  method,  but  no  systematic  development 
or  explanation  of  it.  Leibniz  certainly  did  see  at  least  part  of  this 
tract.  During  the  week  spent  in  London,  he  took  note  of  whatever 
interested  him  among  the  letters  and  papers  of  Collins.  His  memo- 
randa discovered  by  C.  J.  Gerhardt  in  1849  in  the  Hanover  library 
fill  two  sheets.^  The  one  bearing  on  our  question  is  headed  "  Excerpta 
ex  tractatu  Newtoni  Msc.  de  Analysi  per  aequationes  numero  ter- 
minorum  infinitas."  The  notes  are  very  brief,  excepting  those  De 
csolutione  cequaliormm  affectarum,  of  which  there  is  an  almost  com- 
plete copy.  This  part  was  evidently  new  to  him.  If  he  examined 
Newton's  entire  tract,  the  other  parts  did  not  particularly  impress 
him.  From  it  he  seems  to  have  gained  nothing  pertaining  to  the  in- 
finitesimal calculus.  By  the  previous  introduction  of  his  own  al- 
gorithm he  had  made  greater  progress  than  by  what  came  to  his 
knowledge  in  London.  Nothing  mathematical  that  he  had  received 
engaged  his  thoughts  in  the  immediate  future,  for  on  his  way  back 
to  Holland  he  composed  a  lengthy  dialogue  on  mechanical  subjects. 

Fatio  de  Duillier's  insinuations  lighted  up  a  flame  of  discord  which 
a  whole  century  was  hardly  sufficient  to  extinguish.  Leibniz,  who 
had  never  contested  the  priority  of  Newton's  discovery,  and  who 
appeared  to  be  quite  satisfied  with  Newton's  admission  in  his  scholium, 
now  appears  for  the  first  time  in  the  controversy.  He  made  an  ani- 
mated reply  in  the  Acta  eruditorum  and  complained  to  the  Royal 
Society  of  the  injustice  done  him. 

Here  the  affair  rested  for  some  time.  In  the  Quadrature  of  Curves, 
published  1704,  for  the  first  time,  a  formal  exposition  of  the  method 
and  notation  of  fluxions  was  made  public.  In  1705  appeared  an  un- 
1  C.  J.  Gerhardt,  "Leibniz  in  London,"  lac.  cil. 


NEWTON  TO  EULER  215 

favorable  review  of  this  in  the  Acta  enuiitorum,  stating  that  Newton 
uses  and  always  has  used  fluxions  for  the  differences  of  Leibniz.  This 
was  considered  by  Newton's  friends  an  imputation  of  plagiarism  on 
the  part  of  their  chief,  but  this  interpretation  was  always  strenuously 
resisted  by  Leibniz.  John  Keill  (1671-1721),  professor  of  astronomy 
at  Oxford,  undertook  with  more  zeal  than  judgment  the  defence  of 
Newton.  In  a  paper  inserted  in  the  Philosophical  Transactions  of 
1708,  he  claimed  that  Newton  was  the  first  inventor  of  fluxions  and 
"that  the  same  calculus  was  afterward  published  by  Leibniz,  the 
name  and  the  mode  of  notation  being  changed."  Leibniz  complained 
to  the  secretary  of  the  Royal  Society  of  bad  treatment  and  requested 
the  interference  of  that  body  to  induce  Keill  to  disavow  the  intention 
of  imputing  fraud.  John  Keill  was  not  made  to  retract  his  accusation; 
on  the  contrary,  was  authorized  by  Newton  and  the  Royal  Society 
to  explain  and  defend  his  statement.  This  he  did  in  a  long  letter. 
Leibniz  thereupon  complained  that  the  charge  was  now  more  open 
than  before,  and  appealed  for  justice  to  the  Royal  Society  and  to 
Newton  himself.  The  Royal  Society,  thus  appealed  to  as  a  judge, 
appointed  a  committee  which  collected  and  reported  upon  a  large 
mass  of  documents — mostly  letters  from  and  to  Newton,  Leibniz, 
Wallis,  Collins,  etc.  This  report,  called  the  Commercium  epistoliciim, 
appeared  in  the  year  17 12  and  again  in  1722  and  1725,  with  a  Recensio 
prefixed,  and  additional  notes  by  Keill.  The  final  conclusion  in  the 
Commercium  cpistolicum  was  that  Newton  was  "the  first  inventor." 
But  this  was  not  to  the  point.  The  question  was  not  whether  Newton 
was  the  first  inventor,  but  whether  Leibniz  had  stolen  the  method. 
The  committee  had  not  formally  ventured  to  assert  their  belief  that 
Leibniz  was  a  plagiarist.  In  the  following  sentence  they  insinuated 
that  Leibniz  did  take  or  might  have  taken,  his  method  from  that  of 
Newton:  "And  we  find  no  mention  of  his  (Leibniz's)  having  any  other 
DiJJerenlial  Method  than  Moutoi's  before  his  Letter  of  21st  of  June, 
1677,  which  was  a  year  after  a  Copy  of  Mr.  Newton's  Letter,  of  the 
loth  of  December,  1672,  had  been  sent  to  Paris  to  be  communicated 
to  him;  and  about  four  years  after  Mr.  Collins  began  to  communicate 
that  Letter  to  his  Correspondents;  in  which  Letter  the  Method  of 
Fluxions  was  sufficiently  describ'd  to  any  intelligent  Person." 

About  1850  it  was  shown  that  what  H.  Oldenburg  sent  to  Leibniz 
was  not  Newton's  letter  of  Dec.  10,  1672,  but  only  e.xcerpts  from  it 
which  omitted  Newton's  method  of  drawing  tangents  and  could  not 
possibly  convey  an  idea  of  fluxions.  Oldenburg's  letter  was  found 
among  the  Leibniz  manuscripts  in  the  Royal  Library  at  Hanover,  and 
was  published  by  C.  J.  Gerhardt  in  1846,  1848,  1849  and  1855,^  and 
again  later. 

'  See  Essays  on  the  Life  and  Work  of  Newlon  by  Auj^custiis  De  Morgan,  edited,  with 
notes  and  appendices,  by  Philip  I",.  B.  Jourdain,  Chicago  and  London,  i()[4.  Jour- 
dain  gives  on  p.  102  the  bibliography  of  the  publications  of  Newton  and  Leibniz. 


2i6  A  HISTORY  OF  MATHEMATICS 

Moreover,  when  J.  Edleston  in  1850  published  the  Correspondence 
of  Sir  Isaac  Newton  and  Professor  Cotes,  it  became  known  that  the 
Royal  Society  in  17 12  had  not  one,  but  two,  parcels  of  Collins.  One 
parcel  contained  letters  of  James  Gregory,  and  Isaac  Newton's  letter 
of  Dec.  10,  1672,  in  full;  the  other  parcel,  which  was  marked  "To 
Leibnitz,  the  14th  of  June,  1676  About  Mr.  Gregories  remains," 
contained  an  abridgment  of  a  part  of  the  contents  of  the  first  parcel, 
with  nothing  but  an  allusion  to  Newton's  method  described  in  his 
letter  of  Dec.  10,  1672.  In  the  Commercium  epistollcum  Newton's 
letter  was  printed  in  full  and  no  mention  was  made  of  the  existence 
of  the  second  parcel  that  was  marked  "To  Leibnitz.  .  .  ."  Thus  the 
Commercium  cpistolicwn  conveyed  the  impression  that  Newton's  un- 
curtailed  letter  of  Dec.  10,  1672,  had  reached  Leibniz  in  which  fluxions 
"was  sufficiently  described  to  any  intelligent  person,"  while  as  a 
matter  of  fact  the  method  is  not  described  at  all  in  the  letter  which 
Leibniz  received. 

Leibniz  protested  only  in  private  letters  against  the  proceeding  of 
the  Royal  Society,  declaring  that  he  would  not  answer  an  argument 
so  weak.  John  Bernoulli,  in  a  letter  to  Leibniz,  which  was  published 
later  in  an  anonymous  tract,  is  as  decidedly  unfair  towards  Newton 
as  the  friends  of  the  latter  had  been  towards  Leibniz.  John  Keill 
replied,  and  then  Newton  and  Leibniz  appear  as  mutual  accusers  in 
several  letters  addressed  to  third  parties.  In  a  letter  dated  April  9, 
1716,  and  sent  to  Antonio  Schinella  Conti  (1677-1749),  an  Italian 
priest  then  residing  in  London,  Leibniz  again  reminded  Newton  of 
the  admission  he  had  made  in  the  scholium,  which  he  was  now  desirous 
of  disavowing;  Leibniz  also  states  that  he  always  believed  Newton, 
but  that,  seeing  him  connive  at  accusations  which  he  must  have 
known  to  be  false,  it  was  natural  that  he  (Leibniz)  should  begin  to 
doubt.  Newton  did  not  reply  to  this  letter,  but  circulated  some  re- 
marks among  his  friends  which  he  published  immediately  after  hearing 
of  the  death  of  Leibniz,  November  14,  17 16.  This  paper  of  Newton 
gives  the  following  explanation  pertaining  to  the  scholium  in  question: 
"He  [Leibniz]  pretends  that  in  my  book  of  principles  I  allowed  him 
the  invention  of  the  calculus  differentialis,  independently  of  my  own ; 
and  that  to  attribute  this  invention  to  myself  is  contrary  to  my 
knowledge  there  avowed.  But  in  the  paragraph  there  referred  unto 
I  do  not  find  one  word  to  this  purpose."  In  the  third  edition  of  the 
Principia,  1726,  Newton  omitted  the  scholium  and  substituted  in  its 
place  another,  in  which  the  name  of  Leibniz  does  not  appear. 

National  pride  and  party  feeling  long  prevented  the  adoption  of 
impartial  opinions  in  England,  but  now  it  is  generally  admitted  by 

We  recommend  J.  B.  Biot  and  F.  Lefort's  edition  of  the  Commercium  epistoUcumy 
Paris,  1856,  which  exhibits  all  the  alterations  made  in  the  different  reprints  of  this 
publication  and  reproduces  also  H.  Oldenburg's  letter  to  Leibniz  of  July  26,  167O, 
and  other  important  documents  bearing  on  the  controversy. 


NEWTON  TO  EULEk  217 

nearly  all  familiar  with  the  matter,  that  Leibniz  really  was  an  inde- 
pendent inventor.  Perhaps  the  most  telling  evidence  to  show  that 
Leibniz  was  an  independent  inventor  is  found  in  the  study  of  his 
mathematical  papers  (collected  and  edited  by  C.  J.  Gerhardt,  in  seven 
volumes,  Berlin,  1849-1863),  which  point  out  a  gradual  and  natural 
evolution  of  the  rules  of  the  calculus  in  his  own  mind.  "There  was 
throughout  the  whole  dispute,"  says  De  Morgan,  "a  confusion  be- 
tween the  knowledge  of  fluxions  or  differentials  and  that  of  a  calculus 
of  fluxions  or  differentials;  that  is,  a  digested  method  with  general 
rules." 

This  controversy  is  to  be  regretted  on  account  of  the  long  and  bitter 
alienation  which  it  produced  between  English  and  Continental 
mathematicians.  It  stopped  almost  completely  all  interchange  of 
ideas  on  scientific  subjects.  The  English  adhered  closely  to  Newton's 
methods  and,  until  about  1S20,  remained,  in  most  cases,  ignorant  of 
the  brilliant  mathematical  discoveries  that  were  being  made  on  the 
Continent.  The  loss  in  point  of  scientific  advantage  was  almost 
entirely  on  the  side  of  Britain.  The  only  way  in  which  this  dispute 
may  be  said,  in  a  small  measure,  to  have  furthered  the  progress  of 
mathematics,  is  through  the  challenge  problems  by  which  each  side 
attempted  to  annoy  its  adversaries. 

The  recurring  practice  of  issuing  challenge  problems  was  inaugurated 
at  this  time  by  Leibniz.  They  were,  at  first,  not  intended  as  defiances, 
but  merely  as  exercises  in  the  new  calculus.  Such  was  the  problem 
of  the  isochronous  curve  (to  find  the  curve  along  which  a  body  falls 
with  uniform  velocity),  proposed  by  him  to  the  Cartesians  in  1687,  and 
solved  by  Jakob  Bernoulli,  himself,  and  Johann  Bernoulli.  Jakob  Ber- 
noulli proposed  in  the  Acta  eriiditorum  of  1690  the  question  to  find  the 
curve  (the  catenary)  formed  by  a  chain  of  uniform  weight  suspended 
freely  from  its  ends.  It  was  resolved  by  C.  Huygens,  G.  W.  Leibniz, 
Johann  Bernoulli,  and  Jakob  Bernoulli  himself;  the  properties  of  the 
catenary  were  worked  out  methodically  by  David  Gregory  ^  of  Oxford 
and  himself.  In  1696  Johann  Bernoulli  challenged  the  best  mathemati- 
cians in  Europe  to  solve  the  diflScult  problem,  to  find  the  curve  (the 
cycloid)  along  which  a  body  falls  from  one  point  to  another  in  the 
shortest  possible  time.  Leibniz  solved  it  the  day  he  received  it. 
Newton,  de  I'Hospital,  and  the  two  Bernoullis  gave  solutions.  New- 
ton's appeared  anonymously  in  the  Philosophical  Transactions,  but 
Johann  Bernoulli  recognized  in  it  his  powerful  mind,  "tanquam,"  he 
says,  "ex  ungue  leonem."  The  problem  of  orthogonal  trajectories  (a 
system  of  curves  described  by  a  known  law  being  given,  to  describe 
a  curve  which  shall  cut  them  all  at  right  angles)  was  proposed  by 
Johann  Bernoulli  in  a  letter  to  G.  W.  Leibniz  in  1694.  Later  it  was 
long  printed  in  the  Acta  eniditorum,  but  failed  at  first  to  receive  much 

1  Fhll.  Trans.,  London,  1O97. 


2i8  A  HISTORY  OF  MATHEMATICS 

attention.  It  was  again  proposed  in  1 716  by  Leibniz,  to  feel  the  pulse 
of  the  English  mathematicians. 

This  may  be  considered  as  the  first  defiance  problem  professedly 
aimed  at  the  English.  Newton  solved  it  the  same  evening  on  which 
it  was  delivered  to  him,  although  he  was  much  fatigued  by  the  day's 
work  at  the  mint.  His  solution,  as  published,  was  a  general  plan  of 
an  investigation  rather  than  an  actual  solution,  and  was,  on  that 
account,  criticised  by  Johann  BernoulU  as  being  of  no  value.  Brook 
Taylor  undertook  the  defence  of  it,  but  ended  by  using  very  repre- 
hensible language.  Johann  BernoulH  was  not  to  be  outdone  in  in- 
civility, and  made  a  bitter  reply.  Not  long  afterwards  Taylor  sent 
an  open  defiance  to  Continental  mathematicians  of  a  problem  on  the 
integration  of  a  fluxion  of  complicated  form  which  was  known  to 
very  few  geometers  in  England  and  supposed  to  be  beyond  the  power 
of  their  adversaries.  The  selection  was  injudicious,  for  Johann 
Bernoulli  had  long  before  explained  the  method  of  this  and  similar 
integrations.  It  served  only  to  display  the  skill  and  augment  the 
triumph  of  the  followers  of  Leibniz.  The  last  and  most  unskilful 
challenge  was  by  John  Kcill.  The  problem  was  to  find  the  path  of  a 
projectile  in  a  medium  which  resists  proportionally  to  the  square  of 
the  velocity.  Without  first  making  sure  that  he  himself  could  solve 
it,  Keill  boldly  challenged  Johann  Bernoulli  to  produce  a  solution. 
The  latter  resolved  the  question  in  very  short  time,  not  only  for  a 
resistance  proportional  to  the  square,  but  to  any  power  of  the  velocity. 
Suspecting  the  weakness  of  the  adversary,  he  repeatedly  offered  to 
send  his  solution  to  a  confidential  person  in  London,  provided  Keill 
would  do  the  same.  Keill  never  made  a  reply,  and  Johann  Bernoulli 
abused  him  and  cruelly  exulted  over  him.^ 

The  explanations  of  the  fundamental  principles  of  the  calculus,  as 
given  by  Newton  and  Leibniz,  lacked  clearness  and  rigor.  For  that 
reason  it  met  with  opposition  from  several  quarters.  In  1694  Bernhard 
Nieuwentijt  (1654-17 18)  of  Holland  denied  the  existence  of  differentials 
of  higher  orders  and  objected  to  the  practice  of  neglecting  infinitely 
small  quantities.     These  objections  Leibniz  was  not  able  to  meet 

satisfactorily.    In  his  reply  he  said  the  value  of  —  ^^  geometry  could 

be  expressed  as  the  ratio  of  finite  quantities.  In  the  interpretation 
of  dx  and  dy  Leibniz  vacillated.^  At  one  time  they  appear  in  his 
writings  as  finite  lines;  then  they  are  called  infinitely  small  quantities, 
and  again,  quantitates  inassignabiles ,  which  spring  from  quantitates 
assignabiles  by  the  law  of  continuity.  In  this  last  presentation  Leibniz 
approached  nearest  to  Newton. 

ijohn  Playfair,  "Progress  of  the  Mathematical  and  Physical  Sciences"  in 
Encyclopedia  Brilannka,  7th  Ed.,  continued  in  the  8th  Ed.  by  Sir  John  LesHe. 

2  Consult  G.  Vivanti,  II  concello  d'lnfinitesimo.  Saggio  storico.  Nuova  edizione. 
Napoli,  1901 


NEWTON  TO  EULER  21  v 

m  England  the  principles  of  fluxions  were  boldly  attacked  by 
Bishop  George  Berkeley  (1685-1753),  the  eminent  metaphysician,  in 
a  pubhcation  called  the  Analyst  (1734).  He  argued  with  greatacute- 
ness,  contending,  among  other  things,  that  the  fundamental  idea  of 
supposing  a  fmite  ratio  to  exist  between  terms  absolutely  evanescent — 
"the  ghosts  of  departed  quantities,"  as  he  called  them — was  absurd 
and  unintelligible.  Berkeley  claimed  that  the  second  and  third 
fluxions  were  even  more  mysterious  than  the  first  fluxion.  His  con- 
tention that  no  geometrical  quantity  can  be  exhausted  by  division  is 
in  consonance  with  the  claim  made  by  Zeno  in  his  "dichotomy," 
and  the  claim  that  the  actual  infinite  cannot  be  realized.  Most  modern 
readers  recognize  these  contentions  as  untenable.  Berkeley  declared 
as  axiomatic  a  lemma  involving  the  shifting  of  the  hypothesis:  If  x 
receives  an  increment  /,  where  i  is  expressly  supposed  to  be  some 
quantity,  then  the  increment  of  x'^,  divided  by  i,  is  found  to  be  nx"~  '+ 
n{n  —  i)/2  x"'H+  .  .  .  II  now  you  take  i=o,  the  hypothesis  is  shifted 
and  there  is  a  manifest  sophism  in  retaining  any  result  that  was  ob- 
tained on  the  supposition  that  i  is  not  zero.  Berkeley's  lemma  found 
no  favor  among  EngUsh  mathematicians  until  1S03  when  Robert 
Woodhouse  openly  accepted  it.  The  fact  that  correct  results_  are 
obtained  in  the  differential  calculus  by  incorrect  reasoning  is  explained 
by  Berkeley  on  the  theory  of  "  a  compensation  of  errors."  This  theory 
was  later  advanced  also  by  Lagrange  and  L.  N.  M.  Carnot.  The 
publication  of  Berkeley's  Analyst  was  the  most  spectacular  mathe- 
matical event  of  the  eighteenth  century  in  England.  Practically  all 
British  discussions  of  fluxional  concepts  of  that  time  involve  issues 
raised  by  Berkeley.  Berkeley's  object  in  writing  the  Analyst  was  to 
show  that  the  principles  of  fluxions  are  no  clearer  than  those  of  Chris- 
tianity. He  referred  to  an  "  infidel  mithematician"  (Edmund  Halley), 
of  whom  the  story  is  told  ^  that,  when  he  jested  concerning  theological 
questions,  he  was  repulsed  by  Newton  with  the  remark,  "I  have 
studied  these  things;  you  have  not."  A  friend  of  Berkeley,  when  on  a 
bed  of  sickness,  refused  spiritual  consolation,  because  the  great 
mathematician  Halley  had  convinced  him  of  the  inconceivability  of 
the  doctrines  of  Christianity.  This  induced  Berkeley  to  write  the 
Analyst. 

Replies  to  the  Analyst  were  published  by  James  Jurin  (1684-1750) 
of  Trinity  College,  Cambridge  under  the  pseudonym  of  "Philalethes 
Cantabrigiensis"  and  by  John  Walton  of  Dublin.  There  followed 
several  rejoinders.  Jurin's  defence  of  Newton's  fluxions  did  not  meet 
the  approval  of  the  mathematician,  Benjamin  Robins  (1707-1751). 
In  a  Journal,  called  the  Republick  of  Letters  (London)  and  later  in 
the  Works  of  the  Learned,  a  long  and  acrimonious  controversy  was 
carried  on  between  Jurin  and  Robins,  and  later  between  Jurin  and 
Henry  Pcmberton  (1694-1771),  the  editor  of  the  third  edition  of 
1  M;ich  Mechanics,  1907,  pp.  448-449. 


220  A  HISTORY  OF  MATHEMATICS 

Newton's  Principia.  The  question  at  issue  was  the  precise  meaning 
of  certain  passages  in  the  writings  of  Newton:  Did  Newton  hold  that 
there  are  variables  which  reach  their  limits?  Jurin  answered  "Yes"; 
Robins  and  Pemberton  answered  "No."  The  debate  between  Jurin 
and  Robins  is  important  in  the  history  of  the  theory  of  limits.  Though 
holding  a  narrow  view  of  the  concept  of  a  limit  Robins  deserves  credit 
for  rejecting  all  infinitely  small  quantities  and  giving  a  logically  quite 
coherent  presentation  of  fluxions  in  a  pamphlet,  called  A  Discourse 
concerning  the  Nature  and  Certainty  of  Sir  Isaac  Newton's  Methods  of 
Fluxions,  1735.  This  and  Maclaurin's  Fluxions,  1742,  mark  the  top- 
notch  of  mathematical  rigor,  reached  during  the  eighteenth  century 
in  the  exposition  of  the  calculus.  Both  before  and  after  the  period 
of  eight  years,  1S34-1842,  there  existed  during  the  eighteenth  century 
in  Great  Britain  a  mixture  of  Continental  and  British  conceptions  of 
the  new  calculus,  a  superposition  of  British  symbols  and  phraseology 
upon  the  older  Continental  concepts.  Newton's  notation  was  poor  and 
Leibniz's  philosophy  of  the  calculus  was  poor.  The  mixture  repre- 
sented the  temporary  survival  of  the  least  fit  of  both  systems.  The 
subsequent  course  of  events  was  a  superposition  of  the  Leibnizian 
notation  and  phraseology  upon  the  limit-concept  as  developed  by 
Newton,  Jurin,  Robins,  Maclaurin,  D'Alembert  and  later  writers. 

In  France  Michel  Rolle  for  a  time  rejected  the  differential  calculus 
and  had  a  controversy  with  P.  Varignon  on  the  subject.  Perhaps  the 
most  powerful  argument  in  favor  of  the  new  calculus  was  the  con- 
sistency of  the  results  to  which  it  led.  Famous  is  D'Alemhert's  advice 
to  young  students:  "  Allez  en  avant,  et  la  foi  vous  viendra." 

Among  the  most  vigorous  promoters  of  the  calculus  on  the  Conti- 
nent were  the  BernouUis.  They  and  Euler  made  Basel  in  Switzerland 
famous  as  the  cradle  of  great  mathematicians.  The  family  of  Ber- 
nouUis furnished  in  course  of  a  century  eight  members  who  distin- 
guished themselves  in  mathematics.  We  subjoin  the  following  genea- 
logical table: — 

Nicolaus  Bernoulli,  the  Father 

Jakob,  1 654-1 705    Nicolaus  Johann,  1 667-1 748 

Nicolaus,  16S7-1759     Nicolaus,  1695-1726 
Daniel,  1700-1782 
Johann,  17 10-1790 


Daniel    Johann,  1 744-1807    Jakob,  1 759-1 789 

Most  celebrated  were  the  two  brothers  Jakob  (James)  and  Johann 
(John),  and  Daniel,  the  son  of  John.  Jakob  and  Johann  were  staunch 
friends  of  G.  W.  Leibniz  and  worked  hand  in  hand  with  him.  Jakob 
(James)  Bernoulli  (1654-1705)  was  born  in  Basel.    Becoming  inter- 


NEWTON  TO  EULER  221 

csted  in  the  calculus,  he  mastered  it  without  aid  from  a  teacher. 
From  1687  until  his  death  he  occupied  the  mathematical  chair  at  the 
University  of  Basel.  He  was  the  first  to  give  a  solution  to  Leibniz's 
problem  of  the  isochronous  curve.  In  his  solution,  published  in  the 
ylcia  eruditorum,  1690,  we  meet  for  the  first  time  with  the  word 
integral.  Leibniz  had  called  the  integral  calculus  calculus  summalonus, 
but  in  1696  the  term  calculus  intcgralis  was  agreed  upon  between 
Leibniz  and  Johann  Bernoulli.  Jakob  Bernoulli  gave  in  1694  in  the 
Acta  eruditorum  the  formula  for  the  radius  of  curvature  in  rectangular 
co-ordinates.  At  the  same  time  he  gave  the  formula  also  in  polar  co- 
ordinates. He  was  one  of  the  first  to  use  polar  co-ordinates  in  a  gen- 
eral manner  and  not  simply  for  spiral  shaped  curves.^  Jakob  proposed 
the  problem  of  the  catenary,  then  proved  the  correctness  of  Leibniz's 
construction  of  this  curve,  and  solved  the  more  complicated  problems, 
supposing  the  string  to  be  (i)  of  variably  density,  (2)  extensible, 
(3)  acted  upon  at  each  point  by  a  force  directed  to  a  fixed  centre.  Of 
these  problems  he  published  answers  without  explanations,  while  his 
brother  Johann  gave  in  addition  their  theory.  He  determined  the 
shape  of  the  "elastic  curve"  formed  by  an  elastic  plate  or  rod  fixed 
at  one  end  and  bent  by  a  weight  applied  to  the  other  end;  of  the 
"lintearia,"  a  flexible  rectangular  plate  with  two  sides  fixed  hori- 
zontally at  the  same  height,  filled  with  a  Hquid;  of  the  "velaria,"  a 
rectangular  sail  filled  with  wind.  In  the  Acta  eruditorum  of  1694  he 
makes  reference  to  the  lemniscate,  a  curve  which  "formam  refert 
jacentis  not.T  octonarii  00,  seu  complicitiB  in  nodum  fasciae,  sive 
lemnisci."  That  this  curve  is  a  special  case  of  Cassini's  oval  remained 
long  unnoticed  and  was  first  pointed  out  by  Pietro  Ferroni  in  17S2 
and  G.  Saladini  in  1S06.  Jakob  Bernoulli  studied  the  loxodromic  and 
logarithmic  spirals,  in  the  last  of  which  he  took  particular  delight 
from  its  remarkable  property  of  reproducing  itself  under  a  variety  of 
conditions.  Following  the  example  of  Archimedes,  he  willed  that  the 
curve  be  engraved  upon  his  tombstone  with  the  inscription  "eadem 
niutata  resuriw."  In  1696  he  proposed  the  famous  problem  of  isoper- 
imetrical  figures,  and  in  1701  published  his  own  solution.  He  wrote 
a  work  on  Ars  Conjectandi,  published  in  1713,  eight  years  after  his 
death.  It  consists  of  four  parts.  The  first  contains  Huygens'  treatise 
on  probability,  with  a  valuable  commentary.  The  second  part  is  on 
permutations  and  combinations,  which  he  uses  in  a  proof  of  the  bi- 
nomial theorem  for  the  case  of  positive  integral  exponents;  it  contains 
a  formula  for  the  sum  of  the  r^  powers  of  the  first  n  integers,  which  in- 
volves the  so-called  "numbers  of  Bernoulli."  He  could  boast  that 
by  means  of  it  he  calculated  intra  semi-quadrantem  horae  the  sum  of 
the  loth  powers  of  the  first  thousand  integers.  The  third  part  con- 
tains solutions  of  problems  on  probability.  The  fourth  part  is  the 
most  important,  even  though  left  incomplete.  It  contains  "Ber- 
'  G.  Encstrom  in  Bibliotkcca  malhcmatica,  3.  S.,  Vol.  13,  19 12,  p.  76 


222  A  HISTORY  OF  MATHEMATICS 

noulli's  theorem":  If  (r+s)'",  where  the  letters  are  integers  and 
t=r+s,  is  expanded  by  the  binomial  theorem,  then  by  taking  n  large 
enough  the  ratio  of  u  (denoting  the  sum  of  the  greatest  term  and  the 
)j  preceding  terms  and  the  n  followiag  terms)  to  the  sum  of  the  re- 
maining terms  may  be  made  as  great  as  we  please.  Letting  r  and  5 
be  proportional  to  the  probability  of  the  happening  and  failing  of  an 
event  in  a  single  trial,  then  u  corresponds  to  the  probability  that  in  nt 
trials  the  number  of  times  the  event  happens  will  lie  between  n{r~  i) 
and  »(r+i),  both  inclusive.  Bernoulli's  theorem  "will  ensure  him  a 
permanent  place  in  the  history  of  the  theory  of  probability."  ^  Prom- 
inent contemporary  workers  on  probability  were  Montmort  in  France 
and  De  Moivre  in  England.  In  December,  1913,  the  Academy  of 
Sciences  of  Petrograd  celebrated  the  bicentenary  of  the  "law  of  large 
numbers,"  Jakob  Bernoulli's  Ars  conjcdandi  having  been  published 
at  Basel  in  17 13.  Of  his  collected  works,  in  three  volumes,  one  was 
printed  in  17 13,  the  other  two  in  1744. 

Johann  (John)  Bernoulli  (i 667-1 748)  was  initiated  into  mathe- 
matics by  his  brother.  He  afterwards  visited  France,  where  he  met 
Nicolas  Malebranche,  Giovanni  Domenico  Cassini,  P.  de  Lahire,  P. 
Varignon,  and  G.  F.  de  I'Hospital.  For  ten  years  he  occupied  the 
mathematical  chair  at  Groningen  and  then  succeeded  his  brother  at 
Basel.  He  was  one  of  the  most  enthusiastic  teachers  and  most  suc- 
cessful original  investigators  of  his  time.  He  was  a  member  of  almost 
every  learned  society  in  Europe.  His  controversies  were  almost  as 
numerous  as  his  discoveries.  He  was  ardent  in  his  friendships,  but 
unfair,  mean,  and  violent  toward  all  who  incurred  his  dislike — even 
his  own  brother  and  son.  He  had  a  bitter  dispute  with  Jakob  on  the 
isoperimetrical  problem.  Jakob  convicted  him  of  several  paralogisms. 
After  his  brother's  death  he  attempted  to  substitute  a  disguised  solu- 
tion of  the  former  for  an  incorrect  one  of  his  own.  Johann  admired 
the  merits  of  G.  W.  Leibniz  and  L.  Euler,  but  was  blind  to  those  of 
I.  Newton.  He  immensely  enriched  the  integral  calculus  by  his  labors. 
Among  his  discoveries  are  the  exponential  calculus,  the  line  of  swiftest 
descent,  and  its  beautiful  relation  to  the  path  described  by  a  ray 
passing  through  strata  of  variable  density.  In  1694  he  explained  in 
a  letter  to  I'Hospital  the  method  of  evaluating  the  indeterminate 

form  -.    He  treated  trigonometry  by  the  analytical  method,  studied 
o 

caustic  curves  and  trajectories.  Several  times  he  was  given  prizes 
by  the  Academy  of  Science  in  Paris. 

Of  his  sons,  Nicolaus  and  Daniel  were  appointed  professors  of 
mathematics  at  the  same  time  in  the  Academy  of  St.  Petersburg.  The 
former  soon  died  in  the  prime  of  life;  the  latter  returned  to  Basel  in 
1733,  where  he  assumed  the  chair  of  experimental  philosophy.    His 

'  1.  Todhuntcr,  Ilisiory  of  Theor.  of  Prub.,  p.  77. 


NEWTON  TO  EULKR  223 

first  mathematical  publication  was  the  solution  of  a  differential  equa- 
tion proposed  by  J.  F.  Riccati.  He  wrote  a  work  on  hydrodynamics. 
He  was  the  first  to  use  a  suitable  notation  for  inverse  trigonometric 
functions;  in  1729  he  used  AS.  to  represent  arcsine;  L.  Euler  in  1736 
used  At  for  arctangent.'  Daniel  Bernoulli's  investigations  on  prob- 
abihty  are  remarkable  for  their  boldness  and  originality.  He  pro- 
posed the  theory  of  moral  expectation,  which  he  thought  would  give 
results  more  in  accordance  with  our  ordinary  notions  than  the  theory 
of  mathematical  probability.  He  applies  his  moral  expectation  to  the 
so-called  "Petersburg  problem":  A  throws  a  coin  in  the  air;  if  head 
appears  at  the  first  throw  he  is  to  receive  a  shilling  from  B,  if  head 
docs  not  appear  until  the  second  throw  he  is  to  receive  2  shillings,  if 
head  does  not  appear  until  the  third  throw  he  is  to  receive  4  shillings, 
and  so  on:  required  the  expectation  of  A.  By  the  mathematical 
theory,  A's  expectation  is  infinite,  a  paradoxical  result.  A  given  sum 
of  money  not  being  of  equal  importance  to  every  man,  account  should 
be  taken  of  relative  values.  Suppose  A  starts  with  a  sum  a,  then  the 
moral  expectation  in  the  Petersburg  problem  is  finite,  according  to 
Daniel  Bernoulli,  when  a  is  finite;  it  is  2  when  a  =  o,  about  3  when 
a=  10,  about  6  when  a=  1000.^  The  Petersburg  problem  was  discussed 
by  P.  S.  Laplace,  S.  D.  Poisson  and  G.  Cramer.  Daniel  BernouUi's 
''moral  expectation"  has  become  classic,  but  no  one  ever  makes  use 
of  it.  He  applies  the  theory  of  probability  to  insurance;  to  determine 
the  mortality  caused  by  small-pox  at  various  stages  of  life;  to  deter- 
mine the  number  of  survivors  at  a  given  age  from  a  given  number  of 
births;  to  determine  how  much  inoculation  lengthens  the  average 
duration  of  life.  He  showed  how  the  differential  calculus  could  be 
used  in  the  theory  of  probability.  He  and  L.  Euler  enjoyed  the  honor 
of  having  gained  or  shared  no  less  than  ten  prizes  from  the  Academy 
of  Sciences  in  Paris.  Once,  while  travelling  with  a  learned  stranger 
who  asked  his  name,  he  said,  "I  am  Daniel  Bernoulli."  The  stranger 
could  not  believe  that  his  companion  actually  was  that  great  celebrity, 
and  replied  "I  am  Isaac  Newton." 

Johaim  Bernoulli  (born  17 10)  succeeded  his  father  in  the  professor- 
ship of  mathematics  at  Basel.  He  captured  three  prizes  (on  the  cap- 
stan, the  propagation  of  light,  and  the  magnet)  from  the  Academy  of 
Sciences  at  Paris.  Nicolaus  Bernoulli  (born  16S7)  held  for  a  time  the 
mathematical  chair  at  Padua  which  Galileo  had  once  filled.    He  proved 

in  1742  that  7;—=--—.    Johann  Bernoulli  (born  1744)  at  the  age 

of  nineteen  was  appointed  astronomer  royal  at  Berlin,  and  after- 
wards director  of  the  mathematical  department  of  the  Academy.  His 
brother  Jakob  took  upon  himself  the  duties  of  the  chair  of  e.xperi- 

'  G.  Enestrom  in  Bibliothcca  malhematica,  Vol.  6,  pp.  319-321;  Vol.  14,  p.  78. 
- 1.  To'lhuntcr,  Hist,  of  the  Tficor.  of  Proh.,  p.  220. 


224  A  HISTORY  OF  MATHEMATICS 

mental  physics  at  Basel,  previously  performed  by  his  uncle  Jakob, 
and  later  was  appointed  mathematical  professor  in  the  Academy  at 
St.  Petersburg. 

Brief  mention  will  now  be  made  of  some  other  mathematicians 
belonging  to  the  period  of  Newton,  Leibniz,  and  the  elder  BernouUis. 

Guillaume  Francois  Antoine  I'Hospital  (i 661-1704),  a  pupil  of 
Johann  Bernoulli,  has  already  been  mentioned  as  taking  part  in  the 
challenges  issued  by  Leibniz  and  the  BernouUis.  He  helped  power- 
fully in  making  the  calculus  of  Leibniz  better  known  to  the  mass  of 
mathematicians  by  the  publication  of  a  treatise  thereon,  the  Analyse 
des  infiniment  petits,  Paris,  1696.  This  contains  the  method  of  finding 
the  limiting  value  of  a  fraction  whose  two  terms  tend  toward  zero 
at  the  same  time,  due  to  Johann  Bernoulli. 

Another  zealous  French  advocate  of  the  calculus  was  Pierre  Varig- 
non  (1654-1722).  InMcm.  de  Paris,  Annee  MDCCIV,  Paris,  1722,  he 
follows  Ja.  Bernoulli  in  the  use  of  polar  co-ordinates,  p  and  co.  Letting 
x=  p  and  y=lo),  the  equations  thus  changed  represent  wholly  different 
curves.  For  instance,  the  parabolas  .T'"  =  a'""'y  become  Fermatian 
spirals.  Joseph  Saurin  (1659-1737)  solved  the  delicate  problem  of 
how  to  determine  the  tangents  at  the  multiple  points  of  algebraic 
curves.  Francois  Nicole  (1683-175S)  in  171 7  issued  an  elementary 
treatise  on  finite  differences,  in  which  he  finds  the  sums  of  a  consider- 
able number  of  interesting  series.  He  wrote  also  on  roulettes,  particu- 
larly spherical  epicycloids,  and  their  rectification.  Also  interested  in 
finite  differences  was  Pierre  Raymond  de  Montmort  (1678-1719). 
His  chief  writings,  on  the  theory  of  probability,  served  to  stimulate 
his  more  distinguished  successor,  De  Moivre.  Montmort  gave  the 
first  general  solution  of  the  Problem  of  Points.  Jean  Paul  de  Gua 
(1713-1785)  gave  the  demonstration  of  Descartes'  rule  of  signs,  now 
given  in  books.  This  skilful  geometer  wrote  in  1740  a  work  on  analyt- 
ical geometry,  the  object  of  which  was  to  show  that  most  investiga- 
tions on  curves  could  be  carried  on  with  the  analysis  of  Descartes  quite 
as  easily  as  with  the  calculus.  He  shows  how  to  find  the  tangents, 
asymptotes,  and  various  singular  points  of  curves  of  all  degrees,  and 
proves  by  perspective  that  several  of  these  points  can  be  at  infinity. 
Michel  Rolle  (1652-1719)  is  the  author  of  a  theorem  named  after  him. 
That  theorem  is  not  found  in  his  Traite  d'algebre  of  1690,  but  occurs 
in  his  Methode  pour  resoudre  les  egalilez,  Paris,  1691.^  The  name 
"Rolle's  theorem"  was  first  used  by  M.  W.  Drobisch  (1802-1896)  of 
Leipzig  in  1834  and  by  Giusto  Bellavitis  in  1846.  His  Algebre  contains 
his  "method  of  cascades."  In  an  equation  in  v  which  he  has  trans- 
formed so  that  its  signs  become  alternately  plus  and  minus,  he  puts 
v=x+z  and  arranges  the  result  according  to  the  descending  powers 
of  X.    The  coefficients  of  x",  x'^'\  .  .  .,  when  equated  to  zero,  are 

^  See  F.  Cajori  on  the  history  of  Rolle's  Theorem  in  Bibliotlicca  malhcm'itira, 
3rd  S.,  Vol.  JT,  19 1 1,  pp.  300-313. 


NFAVTON  TO  EULER  225 

called  "cascades."  They  are  the  successive  derivatives  of  the  original 
equation  in  v,  each  put  equal  to  zero.  Now  comes  a  theorem  which 
in  modern  version  is:  Between  two  successive  real  roots  oi  f'(v)  =  o 
there  cannot  be  more  than  one  real  root  of  f{v)  =  o.  To  ascertain  the 
root-limits  of  a  given  equation,  RoUe  begins  with  the  cascade  of  lowest 
degree  and  ascends,  solving  each  as  he  proceeds.  This  process  is  very 
laborious. 

Of  Italian  mathematicians,  Riccati  and  Fagnano  must  not  remain 
unmentioned.  Jacopo  Francesco,  Count  Riccati  (1676-1754)  is  best 
known  in  connection  with  his  problem,  called  Riccati's  equation, 
published  in  the  Acta  enuiitorum  in  1724.  He  succeeded  in  integrating 
this  differential  equation  for  some  special  cases.  Long  before  this 
Jakob  Bernoulli  had  made  attempts  to  solve  this  equation,  but  with- 
out success.  A  geometrician  of  remarkable  power  was  Giulio  Carlo, 
Count  de  Fagnano  (1682-1766).  He  discovered  the  following  for- 
mula, Tr=2i  log  — -.,  in  which  he  anticipated  L.  Euler  in  the  use  of 

imaginary  exponents  and  logarithms.  His  studies  on  the  rectification 
of  the  ellipse  and  h}^erbola  are  the  starting-points  of  the  theory  of 
elliptic  functions.  He  showed,  for  instance,  that  two  arcs  of  an  ellipse 
can  be  found  in  an  indefinite  number  of  ways,  whose  difference  is 
expressible  by  a  right  line.  In  the  rectification  of  the  lemniscate  he 
reached  results  which  connect  with  elliptic  functions;  he  showed  that 
its  arc  can  be  divided  geometrically  in  n  equal  parts,  if  n  is  2  -2'", 
3  •  2"',  or  5  •  2'".  He  gave  expert  advice  to  Pope  Benedict  XIV  re- 
garding the  safety  of  the  cupola  of  St.  Peter's  at  Rome.  In  return 
the  Pope  promised  to  publish  his  mathematical  productions.  For 
some  reason  the  promise  was  not  fulfilled  and  they  were  not  published 
until  1750.  Fagnano's  mathematical  works  were  re-published  in  1911 
and  IQ12  by  the  Italian  Society  for  the  Advancement  of  Science. 

In  Germany  the  only  noted  contemporary  of  Leibniz  is  Ehrenfried 
Walter  Tschirnhausen  (i 651-1708),  who  discovered  the  caustic  of 
refiection,  experimented  on  metallic  reflectors  and  large  burning- 
glasses,  and  gave  a  method  of  transforming  equations  named  after 
him.  He  endeavored  to  solve  equations  of  any  degree  by  removing 
all  the  terms  except  the  first  and  last.  This  procedure  had  been  tried 
before  him  by  the  Frenchman  Franqois  Didaurens  and  by  the  Scotch- 
man James  Gregory.^  Gregory's  Vera  circiiU  ct  hyperbolce  quadratura 
(Patavii,  1667)  is  noteworthy  as  containing  a  novel  attempt,  namely, 
to  prove  that  the  quadrature  of  the  circle  cannot  be  effected  by  the 
aid  of  algebra.  His  ideas  were  not  understood  in  his  day,  not  even  by 
C.  Huygens  with  whom  he  had  a  controversy  on  this  subject.  James 
Gregory's  proof  could  not  now  be  considered  binding.  Believing  that 
the  most  simple  methods  (Uke  those  of  the  ancients)  are  the  most 

1  G.  Enestrom  in  Bibliolheca  7nalhcmalica,  3.  S.,  Vol.  9,  190S-9,  pp.  258,  259. 


2  26  A  HISTORY  OF  MATHEMATICS 

correct,  Tschirnhausen  concluded  that  in  the  researches  relating  to 
the  properties  of  curves  the  calculus  might  as  well  be  dispensed  with. 

After  the  death  of  Leibniz  there  was  in  Germany  not  a  single  mathe- 
matician of  note.  Christian  Wolf  (1679-1754),  professor  at  Halle, 
was  ambitious  to  figure  as  successor  of  Leibniz,  but  he  "forced  the 
ingenious  ideas  of  Leibniz  into  a  pedantic  scholasticism,  and  had  the 
unenviable  reputation  of  having  presente'd  the  elements  of  the  arith- 
metic, algebra,  and  analysis  developed  since  the  time  of  the  Renais- 
sance in  the  form  of  Euclid, — of  course  only  in  outward  form,  for  into 
the  spirit  of  them  he  was  quite  unable  to  penetrate"  (H.  Hankel). 

The  contemporaries  and  immediate  successors  of  Newton  in  Great 
Britain  were  men  of  no  mean  merit.  We  have  reference  to  R.  Cotes, 
B.  Taylor,  L.  Maclaurin,  and  A.  de  Moivre.  We  are  told  that  at  the 
death  "of  Roger  Cotes  (1682-1716),  Newton  exclaimed,  "If  Cotes  had 
lived,  we  might  have  known  something."  It  was  at  the  request  of 
Dr.  Bentley  that  R.  Cotes  undertook  the  publication  of  the  second 
edition  of  Newton's  Principia.  His  mathematical  papers  were  pub- 
lished after  his  death  by  Robert  Smith,  his  successor  in  the  Plumbian 
professorship  at  Trinity  College.  The  title  of  the  work,  Harmonia 
Mensurarum,  was  suggested  by  the  following  theorem  contained  in  it: 
If  on  each  radius  vector,  through  a  fixed  point  O,  there  be  taken  a 
point  R,  such  that  the  reciprocal  of  OR  be  the  arithmetic  mean  of  the 
reciprocals  of  0i<:i,0i?2,  •  •  .  Oi?„,  then  the  locus  of  i?  will  be  a  straight 
line.  In  this  work  progress  was  made  in  the  application  of  logarithms 
and  the  properties  of  the  circle  to  the  calculus  of  fluents.  To  Cotes 
we  owe  a  theorem  in  trigonometry  which  depends  on  the  forming  of 
factors  of  x"  —  i.  In  the  Philosophical  Transactions  of  London,  pub- 
Hshed  1 714,  he  develops  an  important  formula,  reprinted  in  his  Har- 
monia Mensurarum,  which  in  modern  notation  is  i  4>=\og  {cos(()+i. 
sin^).)  Usually  this  formula  is  attributed  to  L.  Euler.  Cotes  studied 
the  curve  p"Q=a^,  to  which  he  gave  the  name  "lituus."  Chief  among 
the  admirers  of  Newton  were  B.  Taylor  and  C.  Maclaurin.  The  quar- 
rel between  English  and  Continental  mathematicians  caused  them  to 
work  quite  independently  of  their  great  contemporaries  across  the 
Channel. 

Brook  Taylor  (1685-1731)  was  interested  in  many  branches  of 
learning,  and  in  the  latter  part  of  his  life  engaged  mainly  in  religious 
and  philosophic  speculations.  His  principal  work,  Methodus  incre- 
mentorum  directa  et  inversa,  London,  1715-1717,  added  a  new  branch 
to  mathematics,  now  called  "finite  differences,"  of  which  he  was  the 
inventor.  He  made  many  important  applications  of  it,  particularly 
to  the  study  of  the  form  of  movement  of  vibrating  strings,  the  reduc- 
tion of  which  to  mechanical  principles  was  first  attempted  by  him. 
This  work  contains  also  "Taylor's  theorem,"  and,  as  a  special  case 
of  it,  what  is  now  called  "Maclaurin's  Theorem."  Taylor  discovered 
his  theorem  at  least  three  years  before  its  appearance  in  print.    He 


NEWTON  TO  EULER  227 

gaveitinalettertoJohnMachin,  dated  July  26, 17 12.  Its  importance 
was  not  recognized  by  analysts  for  over  fifty  years,  until  J.  Lagrange 
pointed  out  its  power.  His  proof  of  it  does  not  consider  the  question 
of  convergency,  and  is  quite  worthless.  The  first  more  rigorous  proof 
was  given  a  century  later  by  A.  L.  Cauchy.  Taylor  gave  a  singular 
solution  of  a  ditTerential  equation  and  the  method  of  finding  that 
solution  by  difTerentiation  of  the  differential  equation.  Taylor's 
work  contains  the  first  correct  explanation  of  astronomical  refraction. 
He  wrote  also  a  work  on  linear  perspective,  a  treatise  which,  like  his 
other  writings,  suffers  for  want  of  fulness  and  clearness  of  expression. 
At  the  age  of  twenty-three  he  gave  a  remarkable  solution  of  the  prob- 
lem of  the  centre  of  oscillation,  published  in  17 14.  His  claim  to 
priority  was  unjustly  disputed  by  Johann  Bernoulli.  In  the  Philo-^ 
sophical  Transactions,  Vol.  30,  1717,  Taylor  applies  "Taylor's  series" 
to  the  solution  of  numerical  equations.  He  assumes  that  a  rough 
approximation,  a,  to  a  root  of/(x)=o  has  been  found.  Let/(a)  =  ^, 
f'{a)^k',  f"{a)  =  k",  and  x=a+s.  He  expands  o=f{a+s)  by  his 
"theorem,  discards  all  powers  of  5  above  the  second,  substitutes  the 
values  of  k,  k' ,  k",  and  then  solves  for  s.  By  a  repetition  of  this 
process,  close  approximations  are  secured.  He  makes  the  important 
observation  that  his  method  solves  also  equations  involving  radicals 
and  transcendental  functions.  The  first  application  of  the  Newton- 
Raphson  process  to  the  solution  of  transcendental  equations  was 
made  by  Thomas  Simpson  in  his  Essays  .  .  .  on  Mathematicks, 
London,  1740. 

The  earliest  to  suggest  the  method  of  recurring  series  for  finding 
roots  was  Daniel  Bernoulli  (1700-1782)  who  in  1728  brought  the 
quartic  to  the  form  i  =  ax+bx'+cx^+ex\  then  selected  arbitrarily 
four  numbers  A,  B,  C,  D,  and  a  fifth,  E,  thus,  E=aD+bC+cB+eA, 
also  a  sixth  by  the  same  recursion  formula  F=aE+bD+cC+eB, 
and  so  on.  If  the  last  two  numbers  thus  found  are  M  and  N,  then 
x=M^N  is  an  approximate  root.  Daniel  Bernoulli  gives  no  proof, 
but  is  aware  that  there  is  not  always  convergence  to  the  root.  This 
method  was  perfected  by  Leonhard  Euler  in  his  Introdiiclio  in  analysin 
infinitorum,  1748,  Vol.  I,  Chap.  17,  and  by  Joseph  Lagrange  in  Note 
VI  of  his  Resolution  des  equations  numcriques. 

Brook  Taylor  in  17 17  expressed  a  root  of  a  quadratic  equation  in 
the  form  of  an  infinite  series;  for  the  cubic  Fran(;ois  Nicole  did  simi- 
larly in  1738  and  Clairaut  in  1746.  A.  C.  Clairaut  inserted  the  process 
in  his  Elements  d'algehre.  Thomas  Simpson  determined  roots  by  re- 
version of  series  in  1743  and  by  infinite  series  in  1745.  Marquis  _de 
Courtivron  (1715-1785)  also  expressed  the  roots  in  the  form  of  in- 
finite series,  while  L.  Euler  devoted  several  articles  to  this  topic. ^ 

At  this  time  the  matter  of  convergence  of  the  series  did  not  receive 

'  For  references  see  F.  Cajori,  in  Colorado  College  Publication,  General  Series  51, 
p.  212. 


2  28  A  HISTORY  OF  MATHEMATICS 

proper  attention,  except  in  some  rare  instances.  James  Gregory  of 
Edinburgh,  in  his  Vera  circuli  et  hyperbolcB  quadratura  (1667),  first 
used  the  terms  ''convergent"  and  " divergent "  series,  while  William 
Brouncker  gave  an  argument  which  amounted  to  a  proof  of  the  con- 
vergence of  his  series,  noted  above. 

Colin  Maclaurin  (i  698-1 746)  was  elected  professor  of  mathematics 
at  Aberdeen  at  the  age  of  nineteen  by  competitive  examination,  and 
in  1725  succeeded  James  Gregory  at  the  University  of  Edinburgh. 
He  enjoyed  the  friendship  of  Newton,  and,  inspired  by  Newton's 
discoveries,  he  published  in  17 19  his  Geometria  Organica,  containing 
a  new  and  remarkable  mode  of  generating  conies,  known  by  his  name, 
and  referring  to  the  fact  which  became  known  later  as  ''Cramer's 
paradox,"  that  a  curve  of  the  n^^  order  is  not  always  determined  by 
^n{n+T,)  points,  that  the  number  may  be  less.  A  second  tract,  De 
Linearum  geometricarmn  proprietatibus,  1720,  is  remarkable  for  the 
elegance  of  its  demonstrations.  It  is  based  upon  two  theorems:  the 
first  is  the  theorem  of  Cotes;  the  second  is  Maclaurin's:  If  through 
any  point  O  a  line  be  drawn  meeting  the  curve  in  n  points,  and  at 
these  points  tangents  be  drawn,  and  if  any  other  line  through  0  cut 
the  curve  in  Ri,  Ro,  etc.,  and  the  system  of  n  tangents  in  ri,  ^2,  etc., 

then  27^^=27^-.     This  and  Cotes'  theorem  are  generalizations  of 

theorems  of  Newton.  Maclaurin  uses  these  in  his  treatment  of  curves 
of  the  second  and  third  degree,  culminating  in  the  remarkable  theorem 
that  if  a  quadrangle  has  its  vertices  and  the  two  points  of  intersection 
of  its  opposite  sides  upon  a  curve  of  the  third  degree,  then  the  tangents 
drawn  at  two  opposite  vertices  cut  each  other  on  the  curve.  He  de- 
duced independently  B.  Pascal's  theorem  on  the  hexagram.  Some 
of  his  geometrical  results  were  reached  independently  by  William 
Braikenridge  (about  1700 — after  1759),  a  clergyman  in  Edinburgh. 
The  following  is  known  as  the  " Braikenridge-Maclaurin  theorem": 
If  the  sides  of  a  polygon  are  restricted  to  pass  through  fixed  points 
while  all  the  vertices  but  one  lie  on  fixed  straight  lines,  the  free  vertex 
describes  a  conic  section  or  a  straight  fine.  Maclaurin's  more  general 
statement  {Phil.  Trans.,  1735)  is  thus:  If  a  polygon  move  so  that  each 
of  its  sides  passes  through  a  fixed  point,  and  if  all  its  summits  except 
one  describe  curves  of  the  degrees  m,  n,  p,  etc.,  respectively,  then  the 
free  summit  moves  on  a  curve  of  the  degree  2  mnp  .  .  .,  which  reduces 
to  mnp  .  .  .  when  the  fixed  points  all  lie  on  a  straight  line.  Mac- 
laurin was  the  first  to  write  on  "pedal  curves,"  a  name  due  to  Olry 
Terquem  (1782-1862).  Maclaurin  is  the  author  of  an  Algebra.  The 
object  of  his  treatise  on  Fluxions  was  to  foi:nd  the  doctrine  of  fluxions 
on  geometric  demonstrations  after  the  manner  of  the  ancients,  and 
thus,  by  rigorous  exposition,  answer  such  attacks  as  Berkeley's  that 
the  doctrine  rested  on  false  reasoning.    The  Fluxions  contained  foi 


NEWTON    rO  EULER 


229 


the  first  lime  the  correct  way  of  distinguishing  between  maxima  and 
minima,  and  explained  their  use  in  the  theory  of  multiple  points. 
"Maclaurin's  theorem"  was  previously  given  by  B.  Taylor  and  James 
Stirling,  and  is  but  a  particular  case  of  "Taylor's  theorem."  Maclaurin 
invented  the  trisectrix,  x{x^+y'^)  =  a{y^- ^x-),  which  is  akin  to  the 
Folium  of  Descartes.  Appended  to  the  treatise  on  Fluxions  is  the 
solution  of  a  number  of  beautiful  geometric,  mechanical,  and  as- 
tronomical problems,  in  which  he  employs  ancient  methods  with 
such  consummate  skill  as  to  induce  A.  C.  Clairaut  to  abandon  analytic 
methods  and  to  attack  tlie  problem  of  the  figure  of  the  earth  by  pure 
geometry.  His  solutions  commanded  the  UveUest  admiration  of  J. 
Lagrange.  JSIaclaurin  investigated  the  attraction  of  the  ellipsoid  of 
revolution,  and  showed  that  a  homogeneous  liquid  mass  revolving 
uniformly  around  an  axis  under  the  action  of  gravity  must  assume 
the  form  of  an  elUpsoid  of  revolution.  Newton  had  given  this  theorem 
without  proof.  Notwithstanding  the  genius  of  Maclaurin,  his  in- 
fluence on  the  progress  of  mathematics  in  Great  Britain  was  unfortu- 
nate; for,  by  his  example,  he  induced  his  countrymen  to  neglect 
analysis  and  to  be  indifferent  to  the  wonderful  progress  in  the  higher 
analysis  made  on  the  Continent. 

James  Stirling  (1692-1770),  whom  we  have  mentioned  in  connec- 
tion with  C.  Maclaurin's  theorem  and  Newton's  enumeration  of  72 
forms  of  cubic  curves  (to  which  Stirling  added  4  forms),  was  educated  at 
Glasgow  and  Oxford.  He  was  expelled  from  Oxford  for  corresponding 
with  Jacobites.  For  ten  years  he  studied  in  Venice.  He  enjoyed  the 
friendship  of  Newton.    His  Methodus  differential  is  appeared  in  1730. 

It  remains  for  us  to  speak  of  Abraham,  de  Moivre  (1667-1754), 
who  was  of  French  descent,  but  was  compelled  to  leave  France  at 
the  age  of  eighteen,  on  the  Revocation  of  the  Edict  of  Nantes.  He 
settled  in  London,  where  he  gave  lessons  in  mathematics.  He  ranked 
high  as  a  mathematician.  Newton  himself,  in  the  later  years  of  his 
life,  used  to  reply  to  inquirers  respecting  mathematics,  even  respecting 
his  Principia:  "Go  to  Mr.  De  Moivre;  he  knows  these  things  better 
than  I  do."  He  lived  to  the  advanced  age  of  eighty-seven  and  sank 
into  a  state  of  almost  total  lethargy.  His  subsistence  was  latterly 
dependent  on  the  solution  of  questions  on  games  of  chance  and 
problems  on  probabilities,  which  he  was  in  the  habit  of  giving  at  a 
tavern  in  St.  Martin's  Lane.  Shortly  before  his  death  he  declared 
that  it  was  necessary  for  him  to  sleep  ten  or  twenty  minutes  longer 
every  day.  The  day  after  he  had  reached  the  total  of  over  twenty- 
three  hours,  he  slept  exactly  twenty-four  hours  and  then  passed  away 
in  his  sleep.  De  Moivre  enjoj^ed  the  friendship  of  Newton  and  Halley. 
His  power  as  a  mathematician  lay  in  analytic  rather  than  geometric 
investigation.  He  revolutionized  higher  trigonometry  by  the  dis- 
covery of  the  theorem  known  by  his  name  and  by  extending  the 
tlieorems  on  the  multiplication  and  division  of  sectors  from  the  circle 


i3o  A  HISTORY  OF  MATHEMATICS 

to  the  hyperbola.  His  work  on  the  theory  of  probabihty  surpasses 
anything  done  by  any  other  mathematician  except  P.  S.  Laplace. 
His  principal  contributions  are  his  investigations  respecting  the 
Duration  of  Play,  his  Theory  of  Recurring  Series,  and  his  extension 
of  the  value  of  Daniel  Bernoulli's  theorem  by  the  aid  of  Stirling's 
theorem.^  His  chief  works  are  the  Doctrine  of  Chances,  1716,  the 
Miscellanea  Analytica,  1730,  and  his  papers  in  the  Philosophical 
Transactions.  Unfortunately  he  did  not  publish  the  proofs  of  his 
results  in  the  doctrine  of  chances,  and  J.  Lagrange  more  than  fifty 
years  later  found  a  good  exercise  for  his  skill  in  supplying  the  proofs. 
A  generalization  of  a  problem  first  stated  by  C.  Huygens  has  re- 
ceived the  name  of  "De  Moivre's  Problem:"  Given  n  dice,  each 
having  /  faces,  determine  the  chances  of  throwing  any  given  number 
of  points.  It  was  solved  by  A.  de  Moivre,  P.  R.  de  Montmort,  P.  S. 
Laplace  and  others.  De  Moivre  also  generalized  the  Problem  on  the 
Duration  of  Play,  so  that  it  reads  as  follows:  Suppose  A  has  m  counters, 
and  B  has  n  counters;  let  their  chances  of  winning  in  a  single  game  be 
as  a  to  b;  the  loser  in  each  game  is  to  give  a  counter  to  his  adversary: 
required  the  probability  that  when  or  before  a  certain  number  of  games 
has  been  played,  one  of  the  players  will  have  won  all  the  counters  of 
his  adversary.  De  Moivre's  solution  of  this  problem  constitutes  his 
most  substantial  achievement  in  the  theory  of  chances.  He  employed 
in  his  researches  the  method  of  ordinary  finite  differences,  or  as  he 
called  it,  the  method  of  recurrent  series. 

A  famous  theory  involving  the  notion  of  inverse  probability  was 
advanced  by  Thomas  Bayes.  It  was  published  in  the  London  Philo- 
sophical Transactions,  Vols.  53  and  54  for  the  years  1763  and  1764, 
after  the  death  of  Bayes,  which  occurred  in  1761.  These  researches 
originated  the  discussion  of  the  probabilities  of  causes  as  inferred 
from  observed  effects,  a  subject  developed  more  fully  by  P.  S.  Laplace. 
Using  modern  symbols,  Bayes'  fundamental  theorem  may  be  stated 
thus:  ^  If  an  event  has  happened  p  times  and  failed  q  times,  the 
probability  that  its  chance  at  a  single  trial  lies  between  a  and  b  is 

JxP  (i  -x)?  dx-i-  I    xP  (i  — x)9  dx. 
a  Jo 

A  memoir  of  John  Michell  "On  the  probable  Parallax,  and  Magni- 
tude of  the  fixed  Stars"  in  the  London  Philosophical  Transactions, 
Vol.  57  I,  for  the  year  1767,  contains  the  famous  argument  for  the 
existence  of  design  drawn  from  the  fact  of  the  closeness  of  certain 
stars,  like  the  Pleiades.  "We  may  take  the  six  brightest  of  the 
Pleiades,  and,  supposing  the  whole  number  of  those  stars,  which  are 
equal  in  splendor  to  the  faintest  of  these,  to  be  about  1500,  we  shall 

'  I.  Todhunter,  A  History  of  the  Mathematical  Theory  of  Probability,  Cambridge 
xnd  London,  1865,  pp.  135-193. 
2  I.  Todhunter,  op.  cit.,  p.  295. 


EULER,  T.AGRANGE  AND  LAPLACE  231 

find  the  odds  to  be  near  500,000  to  i,  that  no  six  stars,  out  of  that 
number,  scattered  at  random,  in  the  whole  heavens,  would  be  within 
so  small  a  distance  from  each  other,  as  the  Pleiades  are." 

Elder,  Lagrange,  and  Laplace 

In  the  rapid  development  of  mathematics  during  the  eighteenth 
century  the  leading  part  was  taken,  not  by  the  universities,  but  by 
the  academies.  Particularly  prominent  were  the  academies  at  Berlin 
and  Petrograd.  This  fact  is  the  more  singular,  because  at  that  time 
Germany  and  Russia  did  not  produce  great  mathematicians.  The 
academies  received  their  adornment  mainly  from  the  Swiss  and 
French.  It  was  after  the  French  Revolution  that  schools  gained  their 
ascendancy  over  academies. 

During  the  period  from  1730  to  1820  Switzerland  had  her  L.  Euler; 
France,  her  J.  Lagrange,  P.  S.  Laplace,  A.  M.  Legendre,  and  G.  Monge. 
The  mediocrity  of  French  mathematics  which  marked  the  time  of 
Louis  XIV  was  now  followed  by  one  of  the  very  brightest  periods  of 
all  history.  England,  on  the  other  hand,  which  during  the  unpro- 
ductive period  in  France  had  her  Newton,  could  now  boast  of  no  great 
mathematician.  Except  young  Gauss,  Germany  had  no  great  name. 
France  now  waved  the  mathematical  sceptre.  Mathematical  studies 
among  the  English  and  German  people  had  sunk  to  the  lowest  ebb. 
Among  them  the  direction  of  original  research  was  ill  chosen.  The 
former  adhered  with  excessive  partiality  to  ancient  geometrical 
methods;  the  latter  produced  the  combinatorial  school,  which  brought 
forth  nothing  of  great  value. 

The  labors  of  L.  Euler,  J.  Lagrange,  and  P.  S.  Laplace  lay  in  higher 
analysis,  and  this  they  developed  to  a  wonderful  degree.  By  them 
analysis  came  to  be  completely  severed  from  geometry.  During  the 
preceding  period  the  effort  of  mathematicians  not  only  in  England, 
but,  to  some  extent,  even  on  the  continent,  had  been  directed  toward 
the  solution  of  problems  clothed  in  geometric  garb,  and  the  results  of 
calculation  were  usually  reduced  to  geometric  form.  A  change  now 
took  place.  Euler  brought  about  an  emancipation  of  the  analytical 
calculus  from  geometry  and  established  it  as  an  independent  science. 
Lagrange  and  Laplace  scrupulously  adhered  to  this  separation. 
Building  on  the  broad  foundation  laid  for  higher  analysis  and  me- 
chanics by  Newton  and  Leibniz,  Euler,  with  matchless  fertility  of 
mind,  erected  an  elaborate  structure.  There  are  few  great  ideas  pur- 
sued by  succeeding  analysts  which  were  not  suggested  by  L.  Euler, 
or  of  which  he  did  not  share  the  honor  of  invention.  With,  perhaps, 
less  exuberance  of  invention,  but  with  more  comprehensive  genius  and 
profounder  reasoning,  J.  Lagrange  developed  the  infinitesimal  calculus 
and  put  analytical  mechanics  into  the  form  in  which  we  now  know  it. 
P.  S.  Laplace  applied  the  calculus  and  mechanics  to  the  elaboration 


232  A  HISTORY  OF  MATHEMATICS 

of  the  theory  of  universal  gravitation,  and  thus,  largely  extending  and 
supplementing  the  labors  of  Newton,  gave  a  full  analytical  discussion 
of  the  solar  system.  He  also  wrote  an  epoch-marking  work  on  Prob- 
abiUty.  Among  the  analytical  branches  created  during  this  period 
are  the  calculus  of  Variations  by  Euler  and  Lagrange,  Spherical  Har- 
monics by  Legendre  and  Laplace,  and  Elliptic  Integrals  by  Legendre. 

Comparing  the  growth  of  analysis  at  this  time  with  the  growth 
during  the  time  of  K.  F.  Gauss,  A.  L.  Cauchy,  and  recent  mathe- 
maticians, we  observe  an  important  difference.  During  the  former 
period  we  witness  mainly  a  development  with  reference  to  form.  Plac- 
ing almost  implicit  confidence  in  results  of  calculation,  mathemati- 
cians did  not  always  pause  to  discover  rigorous  proofs,  and  were  thus 
led  to  general  propositions,  some  of  which  have  since  been  found  to 
be  true  in  only  special  cases.  The  Combinatorial  School  in  Germany 
carried  this  tendency  to  the  greatest  extreme;  they  worshipped 
formalism  and  paid  no  attention  to  the  actual  contents  of  formulae. 
But  in  recent  times  there  has  been  added  to  the  dexterity  in  the  formal 
treatment  of  problems,  a  much-needed  rigor  of  demonstration.  A 
good  example  of  this  increased  rigor  is  seen  in  the  present  use  of  in- 
finite series  as  compared  to  that  of  Euler,  and  of  Lagrange  in  his  earlier 
works. 

The  ostracism  of  geometry,  brought  about  by  the  master-minds  of 
this  period,  could  not  last  permanently.  Indeed,  a  new  geometric 
school  sprang  into  existence  in  France  before  the  close  of  this  period. 
J.  Lagrange  would  not  permit  a  single  diagram  to  appear  in  his 
Mccanique  analytique,  but  thirteen  years  before  his  death,  G.  Monge 
published  his  epoch-making  Gcomctrie  descriptive. 

Leonhard  Euler  (1707-1783)  was  born  in  Basel.  His  father,  a 
minister,  gave  him  his  first  instruction  in  mathematics  and  then  sent 
him  to  the  University  of  Basel,  where  he  became  a  favorite  pupil  of 
Johann  Bernoulli.  In  his  nineteenth  year  he  composed  a  dissertation 
on  the  masting  of  ships,  which  received  the  second  prize  from  the 
French  Academy  of  Sciences.  When  Johann  Bernoulli's  two  sons, 
Daniel  and  Nicolaus,  went  to  Russia,  they  induced  Catharine  I,  in 
1727,  to  invite  their  friend  L.  Euler  to  St.  Petersburg,  where  Daniel, 
in  1733,  was  assigned  to  the  chair  of  mathematics.  In  1735  the  solving 
of  an  astronomical  problem,  proposed  by  the  Academy,  for  which 
several  eminent  mathematicians  had  demanded  some  months'  time, 
was  achieved  in  three  days  by  Euler  with  aid  of  improved  methods  of 
his  own.  But  the  effort  threw  him  into  a  fever  and  deprived  him  of 
the  use  of  his  right  eye.  With  still  superior  methods  this  same  problem 
was  solved  later  by  K.  F.  Gauss  in  one  hour!  ^  The  despotism  of 
Anne  I  caused  the  gentle  Euler  to  shrink  from  public  affairs  and  to 
devote  all  his  time  to  science.    After  his  call  to  Berlin  by  Frederick  the 

'  W.  Sartorius  Waltershausen,  Gauss,  zum  Gedichlniss,  Leipzig,  1856. 


EULER,  LAGRANGE  AND  LAPLACE  2^^ 

Great  in  1741,  the  queen  of  Prussia,  who  received  him  kindly,  won- 
dered how  so  distinguished  a  scholar  should  be  so  timid  and  reticent. 
Euler  naively  replied,  "  Madam,  it  is  because  I  come  from  a  country 
where,  when  one  speaks,  one  is  hanged."  It  was  on  the  recommenda- 
tion of  D'Alembert  that  Frederick  the  Great  had  invited  Euler  to 
BerHn.  Frederick  was  no  admirer  of  mathematicians  and,  in  a  letter 
to  V  jltaire,  spoke  of  Euler  derisively  as  "  un  gros  cyclope  de  geometre." 
In  1766  Euler  with  difficulty  obtained  permission  to  depart  from  Berlin 
to  accept  a  call  by  Catharine  II  to  St.  Petersburg.  Soon  after  his 
return  to  Russia  he  became  blind,  but  this  did  not  stop  his  wonderful 
literary  productiveness,  which  continued  for  seventeen  years,  until 
the  day  of  his  death.  He  dictated  to  his  servant  his  Anleitung  zur 
Algebra,  1770,  which,  though  purely  elementary,  is  meritorious  as 
one  of  the  earHest  attempts  to  put  the  fundamental  processes  on  a 
sound  basis. 

The  story  goes  that  when  the  French  philosopher  Denis  Diderot 
paid  a  visit  to  the  Russian  Court,  he  conversed  very  freely  and  gave 
the  younger  members  of  the  Court  circle  a  good  deal  of  lively  atheism. 
Thereupon  Diderot  was  informed  that  a  learned  mathematician  was 
in  possession  of  an  algebraical  demonstration  of  the  existence  of  God, 
and  would  give  it  to  him  before  all  the  Court,  if  he  desired  to  hear  it. 
Diderot  consented.  Then  Euler  advanced  toward  Diderot,  and  said 
gravely,  and  in  a  tone  of  perfect  conviction:  Monsieur,  {a+b"')ln=x, 
done  Dieii  existe;  repondez!  Diderot,  to  whom  algebra  was  Hebrew, 
was  embarrassed  and  disconcerted,  while  peals  of  laughter  rose  on  all 
sides.  He  asked  permission  to  return  to  France  at  once,  which  was 
granted.^ 

Euler  was  such  a  prolific  writer  that  only  in  the  present  century 
have  plans  been  brought  to  maturity  for  a  complete  edition  of  his 
works.  In  1909  the  Swiss  Natural  Science  Association  voted  to  publish 
Euler's  works  in  their  original  language.  The  task  is  being  carried  on 
with  the  financial  assistance  of  German,  French,  American  and  other 
mathematical  organizations  and  of  many  individual  donors.  The 
expense  of  publication  will  greatly  exceed  the  original  estimate  of 
400,000  francs,  owing  to  a  mass  of  new  manuscripts  recently  found  in 
Petrograd. 

The  following  are  his  chief  works:  ^  Introductio  in  analysin  in- 
fjnitortim,  1748,  a  work  that  caused  a  revolution  in  analytical  mathe- 
matics, a  subject  which  had  hitherto  never  been  presented  in  so  general 
and  systematic  manner;  Institntiones  calculi  dij'erentialis,  1755,  and 
I nstitiitiones  calculi  integralis,  1768-17 70,  which  were  the  most  com- 
plete and  accurate  works  on  the  calculus  of  that  time,  and  contained 
not  only  a  full  summary  of  everything  then  known  on  this  subject, 

'  From  De  Morgan's  Budget  of  Paradoxes,  2.  Ed.,  Chicago,  1915,  Vol.  II,  p.  4. 
2  See  G.  Enestrom,  Vcrzcichniss  dcr  Schriften  Leonhard  Eiders,  i.  Lieferung,  iqio, 
1  Lieferung,  1913,  Leipzig. 


234  A  HISTORY  OF  MATHEMATICS 

but  also  the  Beta  and  Gamma  Functions  and  other  original  investi- 
gations; Methodus  inveniendi  lineas  curvas  maximi  minimive  proprietate 
gaudenles,  1744,  which,  displaying  an  amount  of  mathematical  genius 
seldom,  rivalled,  contained  his  researches  on  the  calculus  of  variations 
to  the  invention  of  which  Euler  was  led  by  the  study  of  the  researches 
of  Johann  and  Jakob  Bernoulli.  One  of  the  earliest  problems  bearing 
on  this  subject  was  Newton's  solid  of  revolution,  of  least  resistance, 
reduced  by  him  in  1686  to  a  differential  equation.  {Principia,  Bk.  II, 
Sec.  VII,  Prop.  XXXI  v'.  Scholium.)  Johann  Bernoulli's  problem  of 
the  brachistochrone,  solved  by  him  in  1697,  and  by  his  brother  Jakob 
in  the  same  year,  stimulated  Euler.  The  study  of  isoperimetrical 
curves,  the  brachistochrone  in  a  resisting  medium  and  the  theory  of 
geodesies,  previously  treated  by  the  elder  Bernoullis  and  others,  led 
to  the  creation  of  this  new  branch  of  mathematics,  the  Calculus  of 
Variations.  His  method  was  essentially  geometrical,  which  makes 
the  solution  of  the  simpler  problems  very  clear.  Euler's  Theoria 
motuum  planetariim  et  cometarum,  1744,  theoria  motus  Iuiicb,  1753, 
Theoria  motuum  lun<r,  1772,  are  his  chief  works  on  astronomy;  Ses 
lettres  d  une  princesse  d'Allemagne  sur  quelques  sujets  de  Physique  et  de 
Philosophie,  1770,  was  a  work  which  enjoyed  great  popularity. 

We  proceed  to  mention  the  principal  innovations  and  inventions 
of  Euler.  In  his  Introdudio  (1748)  every  "analytical  expression"  in 
X,  i.  e.  every  expression  made  up  of  powers,  logarithms,  trigonometric 
functions,  etc.,  is  called  a  "function"  of  x.  Sometimes  Euler  used 
another  definition  of  "function,"  namely,  the  relation  between  y 
and  X  expressed  in  the  x-y  plane  by  any  curve  drawn  freehand,  "libero 
manus  ductu."  ^  In  modified  form,  these  two  rival  definitions  are 
traceable  in  all  later  history.  Thus  Lagrange  proceeded  on  the  idea 
involved  in  the  first  definition,  Fourier  on  the  idea  involved  in  the 
second. 

Euler  treated  trigonometry  as  a  branch  of  analysis  and  consistently 
treated  trigonometric  values  as  ratios.  The  term  "trigonometric 
function"  was  introduced  in  1770  by  Gcorg  Simon  Kliigel  (i 739-181 2) 
of  Halle,  the  author  of  a  mathematical  dictionary.^  Euler  developed 
and  systematized  the  mode  of  writing  trigonometric  formulas,  taking, 
for  instance,  the  sinus  totus  equal  to  i.  He  simplified  formulas  by 
the  simple  expedient  of  designating  the  angles  of  a  triangle  by  A,  B,  C, 
and  the  opposite  sides  by  a,  b,  c,  respectively.  Only  once  before  have 
we  encountered  this  simple  device.  It  was  used  in  a  pamphlet  pre- 
pared by  Ri.  Rawlinson  at  Oxford  sometime  between  1655  and  1668.^ 
This  notation  was  re-introduced  simultaneously  with  Euler  by  Thomas 
Simpson  in  England.  We  may  add  here  that  in  1734  Euler  used  the 
notation /(x)  to  indicate  "function  of  x,"  that  the  use  of  e  as  the 

^  F.  Klein,  Elementarmathematik  v.  hoh.  Standpimkte  aiis.,  I,  Leipzig,  1908,  p.  43S. 
-  M.  Cantor,  op.  cit.,  Vol.  IV,  1908,  p.  413. 
^  See  F.  Cajori  in  Nature,  Vol.  94,  19 15,  p.  642. 


EULER,  LAGRANGE  AND  LAl'LACT.  235 

symbol  for  the  natural  base  of  logarithms  was  introduced  by  him  in 
1728/  that  in  1750  he  used  .S"  to  denote  the  half-sum  of  the  sides  of  a 
triangle,  that  in  1755  he  introduced  2  to  signify  "summation,"  that 
in  1777  he  used  /  for  V^^,  a  notation  used  later  by  K.  F.  Gauss. 

We  pause  to  remark  that  in  Euler's  time  Thomas  Simpson  (1710- 
1761),  an  able  and  self-taught  English  mathematician,  for  many  years 
professor  at  the  Royal  Military  Academy  at  Woolwich,  and  author  of 
several  text-books,  was  active  in  perfecting  trigonometry  as  a  science. 
His  Trigonometry,  London,  1748,  contains  elegant  proofs  of  two 
formulas  for  plane  triangles,  {a+b) :  c=cos  \{A  -  B) :  sinlC  and  (a-b): 
c=sin  1{A~-B)\  cos^C,  which  have  been  ascribed  to  the  German  as- 
tronomer Karl  Brandan  Mollweide  (i 774-1825),  who  developed  them 
much  later.  The  first  formula  was  given  in  different  notation  by  I. 
Newtgn  in  his  Universal  Arithmetique;  both  formulas  are  given  by 
Fricdrich  Wilhelm  Oppel  in  1746.- 

Euler  laid  down  the  rules  for  the  transformation  of  co-ordinates  in 
space,  gave  a  methodic  analytic  treatment  of  plane  curves  and  of 
surfaces  of  the  second  order.  He  was  the  first  to  discuss  the  equation 
of  the  second  degree  in  three  variables,  and  to  classify  the  surfaces 
represented  by  it.  By  criteria  analogous  to  those  used  in  the  classi- 
fication of  conies  he  obtained  five  species.  He  devised  a  method  of 
solving  biquadratic  equations  by  assuming  x=\/p+\^q+-\/r,  with 
the  hope  that  it  would  lead  him  to  a  general  solution  of  algebraic 
equations.  The  method  of  elimination  by  solving  a  series  of  linear 
equations  (invented  independently  by  E.  Bezout)  and  the  method  of 
elimination  by  symmetric  functions,  are  due  to  him.  Far  reaching 
are  Euler's  researches  on  logarithms.  Euler  defined  logarithms  as 
exponents,^  thus  abandoning  the  old  view  of  logarithms  as  terms  of 
an  arithmetic  series  in  one-to-one  correspondence  with  terms  of  a 
geometric  series.  This  union  between  the  exponential  and  logarithmic 
concepts  had  taken  place  somewhat  earlier.  The  possibility  of  de- 
fining logarithms  as  exponents  had  been  recognized  by  John  Wallis 
in  1685,  by  Johann  Bernoulli  in  1694,  but  not  till  1742  do  we  find  a 
systematic  exposition  of  logarithms,  based  on  this  idea.  It  is  given 
in  the  introduction  to  Gardiner's  Tables  of  Logarithms,  London,  1742. 
This  introduction  is  "collected  wholly  from  the  papers"  of  William 
Jjnes.  Euler's  influence  caused  the  ready  adoption  of  the  new  defini- 
tion. That  this  view  of  logarithms  was  in  every  way  a  step  in  advance 
has  been  doubted  by  some  writers.  Certain  it  is  that  it  involves  in- 
ternal difficulties  of  a  serious  nature.  Euler  threw  a  stream  of  light 
upon  the  subtle  subject  of  the  logarithms  of  negative  and  imaginary 
numbers.     In  1712  and  1713  this  subject  had  been  discussed  in  a 

^  G.  Enestrom,  Bibliotheca  mathematica ,  Vol.  14,  1913-1914,  p.  81. 
2  A.  V.  Braunmiihl,  op.  cit.,  2.  Teil,  1903,  p.  93;  H.  Wieleitner  in  Bibliotheca 
malhcmaiica,  3.  S.,  Vol.  14,  PP-  348,  349- 

«  See.  L.  Euler.  InlroducUo,  1748,  Chap.  VI,  §  102. 


236  A  HISTORY  OF  MATHEMATICS 

correspondence  between  G.  W.  Leibniz  and  Johann  Bernoulli.^  Leib= 
niz  maintained  that  since  a  positive  logarithm  corresponds  to  a  number 
larger  than  unity,  and  a  negative  logarithm  to  a  positive  number  less 
than  unity,  the  logarithm  of  -  i  was  not  really  true,  but  imaginary; 
hence  the  ratio  -i-^i,  having  no  logarithm,  is  itself  imaginary. 
Moreover,  if  there  really  existed  a  logarithm  of  -  i,  then  half  of  it 
would  be  the  logarithm  of  s/^^,  a.  conclusion  which  he  considered 
absurd.  The  statements  of  Leibniz  involve  a  double  use  of  the  term 
imaginary:  (i)  in  the  sense  of  non-existent,  (2)  in  the  sense  of  a  number 
of  the  type  \/^-  Johann  Bernoulli  maintained  that  -i  has  a 
logarithm.  Since  dx:  x= -dx:  -x,  there  results  by  integration 
log  (a-)=log  (~x);  the  logarithmic  curve  y=log  x  has  therefore  two 
branches,  symmetrical  to  the  y=axis,  as  has  the  hyperbola.  The  corre- 
spondence between  Leibniz  and  Johann  Bernoulli  was  first  published 
in  1745.  In  1 7 14  Roger  Cotes  developed  in  the  Philosophical  Trans- 
actions an  important  theorem  which  was  repubhshed  in  his  Ilarmonia 
mensurarum  (1722).  In  modern  notation  it  is  i</)  =  log  {cos  (p+i  sin  4>). 
In  the  exponential  form  it  was  discovered  again  by  Euler  in  1748. 
Cotes  was  aware  of  the  periodicity  of  the  trigonometric  functions. 
Had  he  applied  this  idea  to  his  formula,  he  might  have  anticipated 
Euler  by  many  years  in  showing  that  the  logarithm  of  a  number  has 
an  infinite  number  of  different  values.  A  second  discussion  of  the 
logarithms  of  negative  numbers  took  place  in  a  correspondence  be- 
tween young  Euler  and  his  revered  teacher,  Johann  Bernoulli,  in  the 
years  1727-1731.-  Bernoulli  argued,  as  before^,  that  log  x=log  {  —  x). 
Euler  uncovered  the  difficulties  and  inconsistencies  of  his  own  and 
BernouUi's  views,  without,  at  that  time,  being  able  to  advance  a 
satisfactory  theory.    He  showed  that  Johann  Bernoulli's  expression 

2"  log   (  —  l) 

for  the  area  of  a  circular  sector  becomes  for  a  quadrant , —   ; 

which  is  incompatible  with  Bernoulli's  claim  that  log  (  — i)=o.  Be- 
tween 1 73 1  and  1747  Euler  made  steady  progress  in  the  mastery  of 
relations  involving  imaginaries.  In  a  letter  of  Oct.  18,  1740,  to 
Johann  BernouUi,  he  stated  that  y=2  cos  x  and  y  =  e*'^"'-|-e-^^"'', 

(Py 
were  both  integrals  of  the  differential  equation  -rY^y^-o  and  were 

equal  to  each  other.  Euler  knew  the  corresponding  expression  for 
sin  X.  Both  expressions  are  given  by  him  in  the  Miscellanea  Berolinen- 
sia,  1743,  and  again  in  his  Inlroductio,  1748,  Vol.  I,  104.  He  gave  the 
value  \/--i^~' =  0)2078795763  as  early  as  1746,  in  a  letter  to  Chris- 
tian Goldbach  (i 690-1 764),  but  makes  no  reference  here  to  the  in- 

^  See  F.  Cajori,  "History  of  the  Exponential  and  Logarithmic  Concepts,"  Amer- 
ican  Math.  Monthly,  Vol.  20,  1913,  pp.  39-42. 

2  See  F.  Cajori  in  Am.  Math.  Monthly,  Vol.  20,  1913,  pp.  44-46. 


EULER,  LAGRANGE  AND  LAPLACE  237 

rinitely  many  values  of  this  imaginary  expression.^  The  creative  work 
on  this  topic  appears  to  have  been  done  in  1747.  During  that  yeai 
and  the  year  following  Euler  debated  this  subject  with  D'Alembert 
in  a  correspondence  of  which  only  a  few  letters  of  Euler  are  extant. - 
In  a  letter  of  April  15,  1747,  Euler  disproves  the  conclusion  upheld  by 
D'Alembert,  that  log  (— i)=o,  and  states  his  own  results  indicating 
that  now  he  had  penetrated  the  subject;  log  n  has  an  infinite  number 
of  values  which  are  all  imaginary,  except  when  n  is  a  positive  number, 
in  which  case  one  logarithm  out  of  this  infinite  number  is  real.  On 
Aug.  19,  1747,  he  said  that  he  had  sent  an  article  to  the  Berlin  Acad- 
emy; this  is  no  doubt  the  article  published  in  1862  under  the  title,  Siir 
les  logarithmes  dcs  nonibres  ncgaiifs  et  imaginaires.  The  reason  why 
Euler  did  not  pubhsh  it  at  the  time  when  it  was  written  can  only  be 
conjectured.  Our  guess  is  that  Euler  became  dissatisfied  with  the 
article.  At  any  rate,  he  wrote  a  new  one  in  1749,  De  la  controverse 
entre  Mrs.  Leibnitz  et  Bernoulli  siir  les  logarithmes  negatifs  et  imaginaires. 
In  1747  he  based  the  proof  that  a  number  has  an  infinity  of  logarithms 
on  the  relation  i(p  =  log{cos(p+i  sin(p);  in  1749  on  the  assumption 
%(i4-co)  =  co,  CO  being  infinitely  small.  He  developed  the  theory  of 
logarithms  of  complex  numbers  a  third  time  in  a  paper  of  1749  on 
Recherches  sur  les  racines  imaginaires  dcs  equations.  The  two  papers 
of  1749  were  published  in  1751  in  the  Berlin  Memoirs.  The  latter 
primarily  aims  to  prove  that  every  equation  has  a  root;  it  was  dis- 
cussed in  1799  by  K.  F.  Gauss  in  his  inaugural  dissertation. 

Euler's  papers  were  not  fully  understood  and  did  not  carry  convic- 
tion. D'Alembert  still  felt  that  the  question  was  not  settled,  and  ad- 
vanced arguments  of  metaphysical,  analytical  and  geometrical  nature 
which  shrouded  the  subject  into  denser  haze  and  helped  to  prolong 
the  controversy  to  the  end  of  the  century.  In  1759  Daviet  de  Foncenex 
(i 734-1 799),  a  young  friend  of  J.  Lagrange,  wrote  on  this  subject. 
In  1768  W.  J.  G.  Karsten  (1732-1787),  professor  at  Biitzow,  later  at 
Halle,  wrote  a  long  treatise  which  contains  an  interesting  graphic 
representation  of  imaginary  logarithms.^  The  debate  on  Euler's 
results  was  carried  on  with  much  warmth  by  the  Italian  mathemati- 
cians. 

The  subject  of  infinite  series  received  new  life  from  him.  To  his 
researches  on  series  we  owe  the  creation  of  the  theory  of  definite  in- 
tegrals by  the  development  of  the  so-called  Eulerian  integrals.  He 
warns  his  readers  occasionally  against  the  use  of  divergent  series,  but 
is  nevertheless  very  careless  himself.  The  rigid  treatment  to  which 
infinite  series  are  subjected  now  was  then  undreamed  of.  No  clear 
notions  existed  as  to  what  constitutes  a  convergent  series.    Neither 

'  P.  H.  Fuss,  Corresp.  math,  et  phys.  de  quelgiics  celchres  geometres  du  xviii^"^^ 
siede,  I,  1S43,  p.  383. 

2  See  F.  Cajori,  Am.  Malh.  Monthty,  Vol.  20,  1913,  pp.  76-79. 
'^  Am.  Math.  Monthly,  Vol.  20,  1913,  p.  iii. 


238  A  HISTORY  OF  MATHEMATICS 

G.  W.  Leibniz  nor  Jakob  and  Johann  Bernoulli  had  entertained  any 
serious  doubt  of  the  correctness  of  the  expression  |  =  i  — i+i  — 1+... 
Guido  Grandi  (1671-1742)  of  Pisa  went  so  far  as  to  conclude  from 
this  that  5  =  0+0+0+  ...  In  the  treatment  of  series  Leibniz  ad- 
vanced a  metaphysical  method  of  proof  which  held  sway  over  the 
minds  of  the  elder  BernouUis,  and  even  of  Euler.^  The  tendency  of 
that  reasoning  was  to  justify  results  which  seem  highly  absurd  to 
followers  of  Abel  and  Cauchy.  The  looseness  of  treatment  can 
best  be  seen  from  examples.  The  very  paper  in  which  Euler  cautions 
against  divergent  series  contains  the  proof  that 

.  .  .-Ti+-+i+w+»^+.  .  .  =  0  as  follows: 
w    n 

„  nil  II 

n+n^+.  .  .= ,  I4--H — :.+  ■  ■  •  =  ■ ; 

i—n        n    tr  n-i 

these  added  give  zero.  Euler  has  no  hesitation  to  write  1-3+5  — 7 
+  .  .  .  =  0,  and  no  one  objected  to  such  results  excepting  Nicolaus 
Bernoulli,  the  nephew  of  Johann  and  Jakob.  Strange  to  say,  Euler 
finally  succeeded  in  converting  Nicolaus  Bernoulli  to  his  own  erroneous 
views.  At  the  present  time  it  is  difficult  to  beheve  that  Euler  should 
have  confidently  written  sin  </>  — 2  sin  2(^+3  sin  3(/)-4  sin  4(^+.  .  . 
=  0,  but  such  examples  afiford  striking  illustrations  of  the  want  of 
scientific  basis  of  certain  parts  of  analysis  at  that  time.  Euler's  proof 
of  the  binomial  formula  for  negative  and  fractional  exponents,  which 
was  widely  reproduced  in  elementary  text-books  of  the  nineteenth 
century,  is  faulty.  A  remarkable  development,  due  to  Euler,  is  what 
he  named  the  hypergeometric  series,  the  summation  of  which  he 
observed  to  be  dependent  upon  the  integration  of  a  linear  differential 
equation  of  the  second  order,  but  it  remained  for  K.  F.  Gauss  to  point 
out  that  for  special  values  of  its  letters,  this  series  represented  nearly 
all  functions  then  known. 

Euler  gave  in  1779  a  series  for  arc  tan  x,  different  from  the  series  of 
James  Gregory,  which  he  applied  to  the  formula  7r=20  arc  tan  1  + 
8  arc  tan  W\  used  for  computing  x.  The  series  was  published  in  1798. 
Euler  reached  remarkable  results  on  the  summation  of  the  reciprocal 
powers  of  the  natural  numbers.  In  1736  he  had  found  the  sum  of  the 
reciprocal  squares  to  be  7r2/6,  and  of  the  reciprocal  fourth  powers  to 
be  7r''/90.  In  an  article  of  1743  which  until  recently  has  been  gen- 
erally overlooked,-  Euler  finds  the  sums  of  the  reciprocal  even  powers 
of  the  natural  numbers  up  to  and  including  the  26th  power.  Later 
he  showed  the  connection  of  coefficients  occurring  in  these  sums  with 
the  "Bernoullian  numbers"  due  to  Jakob  Bernoulli. 

Euler  developed  the  calculus  of  finite  differences  in  the  first  chapters 

1  R.  Reiff,  Geschichte  der  UnendUchen  Reihen,  Tubingen,  1889,  p.  68. 

2  P.  Stackel  in  Bibliolheca  mathematica,  3.  S.,  Vol.  8,  1907-8,  pp.  37-60. 


EULER,  LAGRANGE  AND  LAPLACE 


239 


of  his  Institutiones  calculi  differentialis ,  and  then  deduced  the  differen- 
tial calculus  from  it.  He  established  a  theorem  on  homogeneous  func- 
tions, known  by  his  name,  and  contributed  largely  to  the  theory  of 
differential  equations,  a  subject  which  had  received  the  attention  of 
I.  Newton,  G.  W.  Leibniz,  and  the  Bernoullis,  but  was  still  unde- 
veloped. A.  C.  Clairaut,  Alexis  Fontaine  des  Bertins  (1705-1771), 
and  L.  Euler  about  the  same  time  observed  criteria  of  integrability, 
but  Euler  in  addition  showed  how  to  employ  them  to  determine  in- 
tegrating factors.  The  principles  on  which  the  criteria  rested  involved 
some  degree  of  obscurity.  Euler  was  the  first  to  make  a  systematic 
study  of  singular  solutions  of  differential  equations  of  the  first  order. 
In  1736,  1756  and  176S  he  considered  the  two  paradoxes  which  had 
puzzled  A.  C.  Clairaut:  The  first,  that  a  solution  may  be  reached  by 
differentiation  instead  of  integration;  the  second,  that  a  singular 
solution  is  not  contained  in  the  general  solution.  Euler  tried  to  es- 
tablish an  a  priori  rule  for  determining  whether  a  solution  is  contained 
in  the  general  solution  or  not.  Stimulated  by  researches  of  Count 
de  Fagnano  on  elliptic  integrals,  Euler  established  the  celebrated 
addition-theorem  for  these  integrals.  He  invented  a  new  algorithm 
for  continued  fractions,  which  he  employed  in  the  solution  of  the 
indeterminate  equation  ax+by=c.  We  now  know  that  substantially 
the  same  solution  of  this  equation  was  given  1000  years  earlier,  by 
the  Hindus.  Euler  gave  62  pairs  of  amicable  numbers,  of  which  3 
pairs  were  previously  known:  one  pair  had  been  discovered  by  the 
Pythagoreans,  another  by  Fermat  and  a  third  by  Descartes.^  By 
giving  the  factors  of  the  number  2^"+i  when  n  =  $,  he  pointed  out 
that  this  expression  did  not  always  represent  primes,  as  was  supposed 
by  P.  Fermat.  He  first  supplied  the  proof  to  "Fermat's  theorem," 
and  to  a  second  theorem  of  Fermat,  which  states  that  every  prime 
of  the  form  ^n-\-i  is  expressible  as  the  sum  of  two  squares  in  one  and 
only  one  way.  A  third  theorem,  "Fermat's  last  theorem,"  that 
x"-^yn=z'^,  has  no  integral  solution  for  values  of  n  greater  than  2, 
was  proved  by  Euler  to  be  correct  when  n=^  and  ^=3.  Euler  dis- 
covered four  theorems  which  taken  together  make  out  the  great  law 
of  quadratic  reciprocity,  a  law  independently  discovered  by  A.  M. 
Legendre.^ 

In  1737  Euler  showed  that  the  sum  of  the  reciprocals  of  all  prime 
numbers  is  log,  (log,co),  thereby  initiating  a  line  of  research  on  the 
distribution  of  primes  which  is  usually  not  carried  back  further  than 
to  A.  M.  Legendre.^ 

In  1741  he  wrote  on  partitions  of  numbers  ("partitio  numerorum"). 
In  17S2  he  published  a  discussion  of  the  problem  of  36  oflftcers  of  six 
different  grades  and  from  six  different  regiments,  who  are  to  be  placed 

^  See  Bibliolhcca  malhematica,  3.  S.,  Vol.  9,  p.  263;  Vol.  14,  pp.  351-354. 

2  Oswald  Baumgart,  Ueber  das  Quadralische  Rcciprocil'dtsgesclz.     Leipzig,  1885. 

'  G.  Enestrom  in  Bibliolhcca  malhematica,  3.  S.,  Vol.  13,  1912,  p.  81. 


240  A  HISTORY  OF  MATHEMATICS 

in  a  square  in  such  a  way  that  in  each  row  and  column  there  are  six 
officers,  all  of  difTerent  grades  as  well  as  of  different  regiments.  Euler 
thinks  that  no  solution  is  obtainable  when  the  order  of  the  square  is 
of  the  form  2  mod.  4.  Arthur  Cayley  in  1890  reviewed  what  had  been 
written;  P.  A.  MacMahon  solved  it  in  1915.  It  is  called  the  problem 
of  the  "Latin  squares,"  because  Euler,  in  his  notation,  used  "n  lettres 
latines."  Euler  enunciated  and  proved  a  well-known  theorem,  giving 
the  relation  between  the  number  of  vertices,  faces,  and  edges  of  cer- 
tain polyhedra,  which,  however,  was  known  to  R.  Descartes.  The 
powers  of  Euler  were  directed  also  towards  the  fascinating  subject 
of  the  theory  of  probability,  in  which  he  solved  some  difficult 
problems. 

Of  no  little  importance  are  Euler's  labors  in  analytical  mechanics. 
Says  Whewell:  "The  person  who  did  most  to  give  to  analysis  the 
generahty  and  symmetry  which  are  now  its  pride,  was  also  the  person 
who  made  mechanics  analytical;  I  mean  Euler."  ^  He  worked  out 
the  theory  of  the  rotation  of  a  body  around  a  fixed  point,  established 
the  general  equations  of  motion  of  a  free  body,  and  the  general  equation 
of  hydrodynamics.  He  solved  an  immense  number  and  variety  of 
mechanical  problems,  which  arose  in  his  mind  on  all  occasions.  Thus, 
on  reading  Virgil's  Unes,  "The  anchor  drops,  the  rushing  keel  is  staid," 
he  could  not  help  inquiring  what  would  be  the  ship's  motion  in  such 
a  case.  About  the  same  time  as  Daniel  Bernoulli  he  published  the 
Principle  of  the  Conservation  of  Areas  and  defended  the  principle  of 
"least  action,"  advanced  by  P.  Maupertius.  He  wrote  also  on  tides 
and  on  sound. 

Astronomy  owes  to  Euler  the  method  of  the  variation  of  arbitrary 
constants.  By  it  he  attacked  the  problem  of  perturbations,  explain- 
ing, in  case  of  two  planets,  the  secular  variations  of  eccentricities, 
nodes,  etc.  He  was  one  of  the  first  to  take  up  with  success  the  theory 
of  the  moon's  motion  by  giving  approximate  solutions  to  the  "problem 
of  three  bodies."  He  laid  a  sound  basis  for  the  calculation  of  tables 
of  the  moon.  These  researches  on  the  moon's  motion,  which  captured 
two  prizes,  were  carried  on  while  he  was  blind,  with  the  assistance  of 
his  sons  and  two  of  his  pupils.  His  Mechanica  she  moius  scientia 
analytice  exposita,  Vol.  I,  1736,  Vol.  II,  1742,  is,  in  the  language  of 
Lagrange,  "the  first  great  work  in  which  analysis  is  applied  to  the 
science  of  movement." 

Prophetic  was  his  study  of  the  movements  of  the  earth's  pole.  He 
showed  that  if  the  axis  around  which  the  earth  rotates  is  not  coincident 
with  the  axis  of  figure,  the  axis  of  rotation  will  revolve  about  the  axis 
of  figure  in  a  predictable  period.  On  the  assumption  that  the  earth 
is  perfectly  rigid  he  showed  that  the  period  is  305  days.  The  earth 
is  now  known  to  be  elastic.     From  observations  taken  in  1884-5, 

^W.  Whewell,  History  of  the  Inductive  Sciences,  3rd  Ed.,  Vol.  I,  New  York, 
1858,  p.  363. 


EULER,  LAGRANGE  AND  LAPLACE  241 

S.  C.  Chandler  of  Harvard  found  the  period  to  be  428  days.'  For 
an  earth  of  steel  the  time  has  been  computed  to  be  441  days. 

Euler  in  his  I ntrodiidio  in  analysin  (1748)  had  undertaken  a  classi- 
fication of  quartic  curves,  as  had  also  a  mathematician  of  Geneva, 
Gabriel  Cramer  (1704-1752),  in  his  Introduction  a  Vanalyse  des  lignes 
courbes  algebraiques,  Geneva,  1750.  Both  based  their  classifications 
on  the  behavior  of  the  curves  at  infinity,  obtaining  thereby  eight 
classes  which  were  divided  into  a  considerable  number  of  species. 
Another  classification  was  made  by  E.  Waring,  in  his  Mistcllanea 
anaiytica,  1792,  which  yielded  12  main  divisions  and  84551  species. 
These  classifications  rest  upon  ideas  hardly  in  harmony  with  the 
more  recent  projective  methods,  and  have  been  abandoned.  Cramer 
studied  the  quartic  y^—  x^+ay'^+bx"=o  which  later  received  the  at- 
tention of  F.  Moigno  (1840),  Charles  Briot  and  Jean  Claude  Bouquet, 
and  B.  A.  Nievenglowski  (1895),  and  because  of  its  peculiar  form  was 
called  by  the  French  "courbe  du  diable."  Cramer  gave  also  a  classi- 
fication of  quintic  curves. 

Most  of  Euler's  memoirs  are  contained  in  the  transactions  of  the 
Academy  of  Sciences  at  St.  Petersburg,  and  in  those  of  the  Academy 
at  Berhn.  From  1728  to  1783  a  large  portion  of  the  Petropolitan 
transactions  were  filled  by  his  writings.  He  had  engaged  to  furnish 
the  Petersburg  Academy  with  memoirs  in  sufiicient  number  to  enrich 
its  acts  for  twenty  years — a  promise  more  than  fulfilled,  for  down  to 
1818  the  volumes  usually  contained  one  or  more  papers  of  his,  and 
numerous  papers  are  still  unpublished.  His  mode  of  working  was, 
first  to  concentrate  his  powers  upon  a  special  problem,  then  to  solve 
separately  all  problems  growing  out  of  the  first.  No  one  excelled 
him  in  dexterity  of  accommodating  methods  to  special  problems.  It 
is  easy  to  see  that  mathematicians  could  not  long  continue  in  Euler's 
habit  of  writing  and  publishing.  The  material  would  soon  grow  to 
such  enormous  proportions  as  to  be  unmanageable.  We  are  not  sur- 
prised to  see  almost  the  opposite  in  J.  Lagrange,  his  great  successor. 
The  great  Frenchman  delighted  in  the  general  and  abstract,  rather 
than,  like  Euler,  in  the  special  and  concrete.  His  writings  are  con- 
densed and  give  in  a  nutshell  what  Euler  narrates  at  great  length. 

Jean-le-Rond  D'Alembert  (1717-1783)  was  exposed,  when  an  in- 
fant, by  his  mother  in  a  market  by  the  church  of  St.  Jean-le-Rond, 
near  the  Notre-Dame  in  Paris,  from  which  he  derived  his  Christian 
name.  He  was  brought  up  by  the  wife  of  a  poor  glazier.  It  is  said 
that  when  he  began  to  show  signs  of  great  talent,  his  mother  sent  for 
him,  but  received  the  reply,  "You  are  only  my  step-mother;  the 
glazier's  wife  is  my  mother."  His  father  provided  him  with  a  yearly 
income.  D'Alembert  entered  upon  the  study  of  law,  but  such  was  his 
love  for  mathematics,  that  law  was  soon  abandoned.  At  the  age  of 
twenty-four  his  reputation  as  a  mathematician  secured  for  him  ad- 
^  For  details  see  Nature,  Vol.  97,  1916,  p.  530. 


242  A  HISTORY  OF  MATHEMATICS 

mission  to  the  Academy  of  Sciences.  In  1754  he  was  made  permanent 
secretary  of  the  French  Academy.  During  the  last  years  of  his  life 
he  was  mainly  occupied  with  the  great  French  encyclopaedia,  which 
was  begun  by  Denis  Diderot  and  himself.  D'Alembert  declined,  in 
1762,  an  invitation  of  Catharine  II  to  undertake  the  education  of  her 
son.  Frederick  the  Great  pressed  him  to  go  to  BerUn.  He  made  a 
visit,  but  declined  a  permanent  residence  there.  In  1743  appeared 
his  Traite  de  dynafniqtie,  founded  upon  the  important  general  principle 
bearing  his  name:  The  impressed  forces  are  equivalent  to  the  effective 
forces.  D'Alembert's  principle  seems  to  have  been  recognized  before 
him  by  A.  Fontaine,  and  in  some  measure  by  Johann  Bernoulli  and 
I.  Newton.  D'Alembert  gave  it  a  clear  mathematical  form  and  made 
numerous  applications  of  it.  It  enabled  the  laws  of  motion  and  the 
reasonings  depending  on  them  to  be  represented  in  the  most  general 
form,  in  analytical  language.  D'Alembert  applied  it  in  1744  in  a 
treatise  on  the  equilibrium  and  motion  of  fluids,  in  1746  to  a  treatise 
on  the  general  causes  of  winds,  which  obtained  a  prize  from  the  Berlin 
Academy.  In  both  these  treatises,  as  also  in  one  of  1747,  discussing 
the  famous  problem  of  vibrating  chords,  he  was  led  to  partial  differ- 
ential equations.    He  was  a  leader  among  the  pioneers  in  the  study  of 

such  equations.    To  the  equation  — %  =  a"  ~^,  arising  in  the  problem 

ot  5.1"  "" 

of  vibrating  chords,  he  gave  as  the  general  solution, 

y=f{x+at)  +  (l)(x-dt), 

and  showed  that  there  is  only  one  arbitrary  function,  if  y  be  supposed 
to  vanish  for  .'v:=o  and  .v  =  /.  Daniel  Bernoulli,  starting  with  a  par- 
ticular integral  given  by  Brook  Taylor,  showed  that  this  differential 
equation  is  satisfied  by  the  trigonometric  series 

.      TTX  Tft  ,    a     ■      27ra;  2Trt  . 

y=  asm  -j- .  cos  y-f-  p  sm  —j—  •  cos  — r-  -f- . . ., 

and  claimed  this  expression  to  be  the  most  general  solution.  Thus 
Daniel  Bernoulli  was  the  first  to  introduce  "Fourier's  series"  into 
physics.  He  claimed  that  his  solution,  being  compounded  of  an  in- 
finite number  of  tones  and  overtones  of  all  possible  intensities,  was 
a  general  solution  of  the  problem.  Euler  denied  its  generality,  on 
the  ground  that,  if  true,  the  doubtful  conclusion  would  follow  that 
the  above  series  represents  any  arbitrary  function  of  a  variable. 
These  doubts  were  dispelled  by  J.  Fourier.  J.  Lagrange  proceeded  to 
find  the  sum  of  the  above  series,  but  D'Alembert  objected  to  his 
process,  on  the  ground  that  it  involved  divergent  series.^ 

A  most  beautiful  result  reached  by  D'Alembert,  with  aid  of  his 
principle,  was  the  complete  solution  of  the  problem  of  the  precession 
of  the  equinoxes,  which  had  baffled  the  talents  of  the  best  minds. 
'  R.  Reiff,  op.  cil.,  II.  Abschnitt. 


EULER,  LAGRANGE  AND  LAPLACE  243 

He  sent  to  the  French  Academy  in  1747,  on  the  same  day  with  A.  C. 
Clairaut,  a  solution  of  the  problem  of  three  bodies.  This  had  become 
a  question  of  universal  interest  to  mathematicians,  in  which  each 
vied  to  outdo  all  others.  The  problem  of  two  bodies,  requiring  the 
determination  of  their  motion  when  they  attract  each  other  with 
forces  inversely  proportional  to  the  square  of  the  distance  between 
them,  had  been  completely  solved  by  I.  Newton.  The  "problem  of 
three  bodies"  asks  for  the  motion  of  three  bodies  attracting  each 
other  according  to  the  law  of  gravitation.  Thus  far,  the  complete 
solution  of  this  has  transcended  the  power  of  analysis.  The  general 
differential  equations  of  motion  were  stated  by  P.  S.  Laplace,  but 
the  difficulty  arises  in  their  integration.  The  "solutions"  given  at 
that  time  are  merely  convenient  methods  of  approximation  in  special 
cases  when  one  body  is  the  sun,  disturbing  the  motion  of  the  moon 
around  the  earth,  or  where  a  planet  moves  under  the  influence  of  the 
sun  and  another  planet.  The  most  important  eighteenth  century 
researches  on  the  problem  of  three  bodies  are  due  to  J.  Lagrange.  In 
1772  a  prize  was  awarded  him  by  the  Paris  Academy  for  his  Essai 
sur  le  probleme  des  trois  corps.  He  shows  that  a  complete  solution  of 
the  problem  requires  only  that  we  know  every  moment  the  sides  of 
the  triangle  formed  by  the  three  bodies,  the  solution  of  the  triangle 
depending  upon  two  differential  equations  of  the  second  order  and 
one  differential  equation  of  the  third.  He  found  particular  solutions 
when  the  triangles  remain  all  similar. 

In  the  discussion  of  the  meaning  of  negative  quantities,  of  the 
fundamental  processes  of  the  calculus,  of  the  logarithms  of  complex 
numbers,  and  of  the  theory  of  probabiUty,  D'Alembert  paid  some 
attention  to  the  philosophy  of  mathematics.  In  the  calculus  he 
favored  the  theory  of  limits.  He  looked  upon  infinity  as  nothing  but 
a  limit  which  the  finite  approaches  without  ever  reaching  it.  His 
criticisms  were  not  always  happy.  When  students  were  halted  by 
the  logical  diflficulties  of  the  calculus,  D'Alembert  would  say,  "Allez 
en  avant,  et  la  foi  vous  viendra."  He  argued  that  when  the  prob- 
ability of  an  event  is  very  small,  it  ought  to  be  taken  o.  A  coin  is  to 
be  tossed  100  times  and  if  head  appear  at  the  last  trial,  and  not  before, 
A  shall  pay  B  2^*^"  crowns.    By  the  ordinary  theory  B  should  give  A 

1  crown  at  the  start,  which  should  not  be,  argues  D'Alembert,  be- 
cause B  will  certainly  lose.  This  view  was  taken  also  by  Count  de 
Buffon.  D'Alembert  raised  other  objections  to  the  principles  of 
probability. 

The  naturalist,  Conite  de  Buffon  (1707-178S),  wrote  an  Essal 
d' arithmetique  morale,  1777.  In  the  study  of  the  Petersburg  problem, 
he  let  a  child  toss  a  coin  2084  times,  which  produced  10057  crowns; 
there  were  106 1  games  which  produced  i  crown,  494  which  produced 

2  crowns  and  so  on.^    He  was  one  of  the  first  to  emphasize  the  desir- 

'   For  references,  see  I.  Todhunler,  History  of  Theory  of  Probability,  p.  346. 


244  A  HISTORY  OF  MATHEM/.TICS 

ability  of  verifying  the  theory  by  actual  tria\  He  also  introduced 
what  is  called  "local  probability"  by  the  consideration  of  problems 
that  require  the  aid  of  geometry.  Some  studies  along  this  line  had 
been  carried  on  earUer  by  John  Arbuthnot  (11358-1735)  and  Thomas 
Simpson  in  England.  Count  de  Buffon  derived  the  probability  that 
a  needle  dropped  upon  a  plane,  ruled  with  equidistant,  parallel  lines, 
will  fall  across  one  of  the  lines. 

The  probability  of  the  correctness  of  judgments  determined  by  a 
majority  of  votes  was  examined  mathematically  by  J ean-Antoine- 
Nicolas  Caritat  de  Condorcet  (i 743-1 794).  His  general  conclusions 
are  not  of  great  importance;  they  are  that  voters  must  be  enUghtened 
men  in  order  to  ensure  our  confidence  in  their  decisions.^  He  held 
that  capital  punishment  ought  to  be  abohshed,  on  the  ground  that, 
however  large  the  probability  of  the  correctness  of  a  single  decision, 
there  will  be  a  large  probability  that  in  the  course  of  many  decisions 
some  innocent  person  will  be  condemned.^ 

Alexis  Claude  Clairaut  (17 13-1765)  was  a  youthful  prodigy.  He 
read  G.  F.  de  I'Hospital's  works  on  the  infinitesimal  calculus  and  on 
conic  sections  at  the  age  of  ten.  In  1731  was  pubUshed  his  Recherches 
sur  les  courbes  a  double  courbure,  which  he  had  ready  for  the  press 
when  he  was  sixteen.  It  was  a  work  of  remarkable  elegance  and  se- 
cured his  admission  to  the  Academy  of  Sciences  when  still  under  legal 
age.  In  1731  he  gave  a  proof  of  the  theorem  enunciated  by  I.  Newton, 
that  every  cubic  is  a  projection  of  one  of  five  divergent  parabolas. 
Clairaut  formed  the  acquaintance  of  Pierre  Louis  Moreau  de  Mauper- 
tius  (1698-1759),  whom  he  accompanied  on  an  expedition  to  Lapland 
to  measure  the  length  of  a  degree  of  the  meridian.  At  that  time  the 
shape  of  the  earth  was  a  subject  of  serious  disagreement.  I.  Newton 
and  C.  Huygens  had  concluded  from  theory  that  the  earth  was  flat- 
tened at  the  poles.  About  1712  Jean-Dominique  Cassini  (1625-1712) 
and  his  son  Jacques  Cassini  (1677-1756)  measured  an  arc  extending 
from  Dunkirk  to  Perpignan  and  arrived  at  the  startling  result  that 
the  earth  is  elongated  at  the  poles.  To  decide  between  the  conflicting 
opinions,  measurements  were  renewed.  Maupertius  earned  by  his 
work  in  Lapland  the  title  of  "earth  flattener"  by  disproving  the 
Cassinian  tenet  that  the  earth  was  elongated  at  the  poles,  and  showing 
that  Newton  was  right.  On  his  return,  in  1743,  Clairaut  published 
a  work,  Theorie  de  la  figure  de  la  Terre,  which  was  based  on  the  results 
of  C.  Maclaurin  on  homogeneous  ellipsoids.  It  contains  a  remarkable 
theorem,  named  after  Clairaut,  that  the  sum  of  the  fractions  ex- 
pressing the  ellipticity  and  the  increase  of  gravity  at  the  pole  is  equal 
to  2^  times  the  fraction  expressing  the  centrifugal  force  at  the  equator, 
the  unit  of  force  being  represenLed  by  the  force  of  gravity  at  the 
equator.  This  theorem  is  independ  ent  of  any  hypothesis  with  respect 
to  the  law  of  densities  of  the  successive  strata  of  the  earth.  It  em- 
1 1.  Todhunter,  History  of  Theory  of  Prob.,  Chapter  17. 


EULER,  LAGRANGE  AND  LAPLACE  245 

bodies  most  of  Clairaut's  researches.  I.  Todhunter  says  that  "in 
the  figure  of  the  earth  no  other  person  has  accompUshed  so  much  as 
Clairaut,  and  the  subject  remains  at  present  substantially  as  he  left 
it,  though  the  form  is  different.  The  splendid  analysis  which  Laplace 
supplied,  adorned  but  did  not  really  alter  the  theory  which  started 
from  the  creative  hands  of  Clairaut." 

In  1752  he  gained  a  prize  of  the  St.  Petersburg  Academy  for  his 
paper  on  Thcorie  de  la  Luiie,  in  which  for  the  first  time  modern  analysis 
is  applied  to  lunar  motion.  This  contained  the  explanation  of  the 
motion  of  the  lunar  apsides.  This  motion,  left  unexplained  by  L 
Newton,  seemed  to  him  at  first  inexplicable  by  Newton's  law,  and 
he  was  on  the  point  of  advancing  a  new  hypothesis  regarding  gravi- 
tation, when,  taking  the  precaution  to  carry  his  calculation  to  a  higher 
degree  of  approximation,  he  reached  results  agreeing  with  observation. 
The  motion  of  the  moon  was  studied  about  the  same  time  by  L.  Euler 
and  D'Alembert.  Clairaut  predicted  that  "Halley's  Comet,"  then 
expected  to  return,  would  arrive  at  its  nearest  point  to  the  sun  on 
April  13,  1759,  a  date  which  turned  out  to  be  one  month  too  late. 
He  applied  the  process  of  differentiation  to  the  differential  equation 
now  known  by  his  name  and  detected  its  singular  solution.  The  same 
process  had  been  used  earlier  by  Brook  Taylor. 

In  their  scientific  labors  there  was  between  Clairaut  and  D'Alembert 
great  rivalry,  often  far  from  friendly.  The  growing  ambition  of 
Clairaut  to  shine  in  society,  where  he  was  a  great  favorite,  hindered 
his  scientific  work  in  the  latter  part  of  his  life. 

The  astronomer  Jean-Dominique  Cassini,  whom  we  mentioned 
above,  is  the  inventor  of  a  quartic  curve  which  was  published  in  his 
son's  Elements  (Tastronomie,  1749.  The  curve  bears  the  name  of 
"Cassini's  oval"  or  "general  lemniscate."  It  grew  out  of  the  study 
of  a  problem  in  astronomy.^    Its  equation  is  (x^-t-/) 2— 2a- (.»-—/) + 

Johann  Heinrich  Lambert  (1728-1777),  born  at  Miihlhausen  in 
Alsace,  was  the  son  of  a  poor  tailor.  While  working  at  his  father's 
trade,  he  acquired  through  his  own  unaided  efforts  a  knowledge  of 
elementary  mathematics.  At  the  age  of  thirty  he  became  tutor  in  a 
Swiss  family  and  secured  leisure  to  continue  his  studies.  In  his 
travels  with  his  pupils  through  Europe  he  became  acquainted  with 
the  leading  mathematicians.  In  1764  he  settled  in  Berlin,  where  he 
became  member  of  the  Academy,  and  enjoyed  the  society  of  L.  Euler 
and  J.  Lagrange.  He  received  a  small  pension,  and  later  became 
editor  of  the  Berlin  Epiiemeris.  His  many-sided  scholarship  reminds 
one  of  Leibniz.  It  cannot  be  said  that  he  was  overburdened  with 
modesty.  When  Frederick  the  Great  asked  him  in  their  first  inter- 
view, which  science  he  was  most  proficient  in,  he  replied  curtly,  "All." 

1  G.  Loria,  Ehenc  Ctirven  (F.  Schutte),  T,  19 lo,  p.  208. 


^46  A  HISTORY  OF  MATHEMATICS 

To  the  emperor's  further  question,  how  he  attained  this  mastery,  he 
said,  "Like  the  celebrated  Pascal,  by  my  own  self." 

In  his  Cosniological  Letters  he  made  some  remarkable  prophecies 
regarding  the  stellar  system.  He  entered  upon  plans  for  a  mathe- 
matical symbolic  logic  of  the  nature  once  outlined  by  G.  W.  Leibniz. 
In  mathematics  he  made  several  discoveries  which  were  extended 
and  overshadowed  by  his  great  contemporaries.  His  first  research 
on  pure  mathematics  developed  in  an  infinite  series  the  root  x  of  the 
equation  x"'--\-px  =  q.  Since  each  equation  of  the  form  ax^+bx?=d 
can  be  reduced  to  x^''-]-px=q  in  two  ways,  one  or  the  other  of  the  two 
resulting  series  was  always  found  to  be  convergent,  and  to  give  a 
value  of  A".  Lambert's  results  stimulated  L.  Euler,  who  extended  the 
method  to  an  equation  of  four  terms,  and  particularly  J.  Lagrange, 
who  found  that  a  function  of  a  root  of  a— x+0(x)  =o  can  be  expressed 
by  the  series  bearing  his  name.  In  1761  Lambert  communicated  to 
the  Berlin  Academy  a  memoir  (published  1768),  in  which  he  proves 
rigorously  that  7r  is  irrational.  It  is  given  in  simplified  form  in  Note  IV 
of  A.  M.  Legendre's  Geometrie,  where  the  proof  is  extended  to  x^. 
Lambert  proved  that  if  x  is  rational,  but  not  zero,  then  neither  e* 
nor  tan  x  can  be  a  rational  number;  since  tan  7r/4=i,  it  follows  that 

—  or  7r  cannot  be  rational.  Lambert's  proofs  rest  on  the  expression 
4 

for  e  as  a  continued  fraction  given  by  L.  Euler  ^  who  in  1737  had  sub- 
stantially shown  the  irrationality  of  e  and  c^.  There  were  at  this 
time  so  many  circle  squarers  that  in  1775  the  Paris  Academy  found  it 
necessary  to  pass  a  resolution  that  no  more  solutions  on  the  quadrature 
of  the  circle  should  be  examined  by  its  officials.  This  resolution  ap- 
plied also  to  solutions  of  the  duplication  of  the  cube  and  the  trisection 
of  an  angle.  The  conviction  had  been  growing  that  the  solution  of 
the  squaring  of  the  circle  was  impossible,  but  an  irrefutable  proof 
was  not  discovered  until  over  a  century  later.  Lambert's  Freye  Per- 
spective, 1759  and  1773,  contains  researches  on  descriptive  geometry, 
and  entitle  him  to  the  honor  of  being  the  forerunner  of  Monge.  In 
his  effort  to  simplify  the  calculation  of  cometary  orbits,  he  was  led 
geometrically  to  some  remarkable  theorems  on  conies,  for  instance 
this:  "If  in  two  ellipses  having  a  common  major  axis  we  take  two 
such  arcs  that  their  chords  are  equal,  and  that  also  the  sums  of  the 
radii  vectores,  drawn  respectively  from  the  foci  to  the  extremities  of 
these  arcs,  are  equal  to  each  other,  then  the  sectors  formed  in  each 
ellipse  by  the  arc  and  the  two  radii  vectores  are  to  each  other  as  the 
square  roots  of  the  parameters  of  the  ellipses."  ^ 

Lambert  elaborated  the  subject  of  hyperbolic  functions  which  he 
designated  by  sink  x,  cosh  x,  etc.    He  was,  however,  not  the  first  to 

1  R.  C.  Archibald  in  Am.  Malh.  Monthly,  Vol.  21,  1914,  p.  253. 

2  M.  Chr^sles,  Geschichte  der  Geometrie,  1839,  p.  183. 


EULER,  LAGRANGE  AND  LAPLACE  247 

introduce  them  into  trigonometry.  That  honor  falls  upon  Vincenzo 
Riccati  (1707-1775),  a  son  of  Jacopo  Riccati.^ 

In  1770  Lambert  published  a  7-place  table  of  natural  logarithms 
for  numbers  i-ioo.  In  1778  one  of  his  pupils,  Johann  Karl  Schulze, 
published  extensive  tables  which  included  the  48-place  table  of  nat- 
ural logarithms  of  primes  and  many  other  numbers  up  to  10,009, 
which  had  been  computed  by  the  Dutch  artillery  officer,  Wolfram. 
A  feat  even  more  remarkable  than  Wolfram's,  was  the  computation 
of  the  common  logarithms  of  numbers  i-ioo  and  of  all  primes  from 
100  to  1 100,  to  61  places,  by  Abraham  Sharp  of  Yorkshire,  who  was 
some  time  assistant  to  Flamsteed  at  the  English  Royal  Observatory. 
They  were  published  in  Sharp's  Geometry  Improvd,  1717. 

John  Landen  (17 19-1790)  was  an  English  mathematician  whose 
writings  served  as  the  starting-point  of  investigations  by  L.  Euler, 
J.  Lagrange,  and  A.  AI.  Legendre.  Landen's  capital  discovery,  con- 
tained in  a  memoir  of  1755,  was  that  every  arc  of  the  hyperbola  is 
immediately  rectified  by  means  of  two  arcs  of  an  ellipse.  In  his 
"residual  analysis"  he  attempted  to  obviate  the  metaphysical  diffi- 
culties of  fluxions  by  adopting  a  purely  algebraic  method.  J.  La- 
grange's Calciil  des  Fondions  is  based  upon  this  idea.  Landen  showed 
how  the  algebraic  expression  for  the  roots  of  a  cubic  equation  could 
be  derived  by  application  of  the  differential  and  integral  calculus. 
Most  of  the  time  of  this  suggestive  writer  was  spent  in  the  pursuits 
of  active  life. 

Of  influence  in  the  teaching  of  mathematics  in  England  was  Charles 
Hulton  (1737-1823),  for  many  years  professor  at  the  Royal  Military 
Academy  of  Woolwich.  In  17S5  he  published  his  Mathematical  Tables, 
and  in  1795  his  Mathetnatkal  and  Philosophical  Dictionary,  the  best 
work  of  its  kind  that  has  appeared  in  the  English  language.  His 
Elements  of  Conic  Sections,  1789,  is  remarkable  as  being  the  first  work 
in  which  each  equation  is  rendered  conspicuous  by  being  printed  in 
a  separate  line  by  itself.^ 

It  is  well  known  that  the  Newton-Raphson  method  of  approxima- 
tion to  the  roots  of  numerical  equations,  as  it  was  handed  down  from 
the  seventeenth  century,  labored  under  the  defect  of  insecurity  in 
the  process,  so  that  the  successive  corrections  did  not  always  yield 
results  converging  to  the  true  value  of  the  root  sought.  The  removal 
of  this  defect  is  usually  attributed  to  J.  Fourier,  but  he  was  anticipated 
half  a  century  by  /.  Raym.  Mourraille  in  his  Traite  de  la  resolution 
des  equations  en  general,  Marseille  et  Paris,  1768.  Mourraille  was  for 
fourteen  years  secretary  of  the  academy  of  sciences  in  Marseille;  later 
he  became  mayor  of  the  city.  Unlike  I.  Newton  and  J.  Lagrange, 
Mourraille  and  J.  Fourier  introduced  also  geometrical  considerations. 
Mourraille  concluded  that  security  is  insured  if  the  first  approximation 

1  M.  Cantor,  op.  cit.,  Vol.  IV,  1908,  p.  411- 

2  M.  Cantor,  op.  cit.,  Vol.  IV,  1908,  p.  465. 


248  A  HISTORY  OF  MATHEMATICS 

a  is  so  selected  that  the  curve  is  convex  toward  the  axis  of  x  for  the 
interval  between  a  and  the  root.  He  shows  that  this  condition  is 
sufficient,  but  not  necessary.^ 

In  the  eighteenth  century  proofs  were  given  of  Descartes'  Rule  of 
Signs  which  its  discoverer  had  enunciated  without  demonstration. 
G.  W.  Leibniz  had  pointed  out  a  line  of  proof,  but  did  not  actually 
give  it.  In  1675  Jean  Prestd  (1648-1690)  published  at  Paris  in  his 
Elemens  des  mathematiques  a  proof  which  he  afterwards  acknowledged 
to  be  insufficient.  In  1728  Johann  Andreas  Segner  (1704-1777)  pub- 
lished at  Jena  a  correct  proof  for  equations  having  only  real  roots. 
In  1756  he  gave  a  general  demonstration,  based  on  the  consideration 
that  multiplying  a  polynomial  by  (x— a)  increases  the  number  of 
variations  by  at  least  one.  Other  proofs  were  given  by  Jean  Paul  de 
Gua  de  Malves  (1741),  Isaac  Milner  (1778),  Friedrich  Wilhelni  Stiilmer, 
Abraham  Gotthelf  Kastner  (1745),  Edward  Waring  (1782),  /.  A. 
Gruneri  (1827),  A'.  F.  Gauss  (1828J.  Gauss  showed  that,  if  the  num- 
ber of  positive  roots  falls  short  of  the  number  of  variations,  it  does  so 
by  an  even  number.  E.  Laguerre  later  extended  the  rule  to  poly- 
nomials with  fractional  and  incommensurable  exponents,  and  to  in- 
finite series.^  It  was  established  by  De  Gua  de  Malves  that  the 
absence  of  2m  successive  terms  indicates  2m  imaginary  roots,  while 
the  absence  of  2m+i  successive  terms  indicates  2tn+2  or  2w  imagin- 
ary roots,  according  as  the  two  terms  between  which  the  deficiency 
occurs  have  like  or  unlike  signs. 

Edward  Waring  (i  734-1 798)  was  born  in  Shrewsbury,  studied  at 
Magdalene  College,  Cambridge,  was  senior  wrangler  in  1757,  and 
Lucasian  professor  of  mathematics  since  1760.  He  published  Mis- 
cellanea analytica  in  1762,  Meditationes  algebraicce  in  1770,  Proprietatis 
algehraicarum  curvarum  in  1772,  and  Meditationes  analyticcB  in  1776. 
These  works  contain  many  new  results,  but  are  difficult  of  compre- 
hension on  account  of  his  brevity  and  obscurity  of  exposition.  He  is 
said  not  to  have  lectured  at  Cambridge,  his  researches  being  thought 
unsuited  for  presentation  in  the  form  of  lectures.  He  admitted  that 
he  never  heard  of  any  one  in  England,  outside  of  Cambridge,  who  had 
read  and  understood  his  researches. 

In  his  Meditationes  algebraiccB  are  some  new  theorems  on  number. 
Foremost  among  these  is  a  theorem  discovered  by  his  friend  John 
Wilson  (i 741-1793)  and  universally  known  as  "Wilson's  theorem." 
Waring  gives  the  theorem,  known  as  "  Waring's  theorem,"  that  every 
integer  is  either  a  cube  or  the  sum  of  2,  3,  4,  5,  6,  7,  8  or  9  cubes,  either 
a  fourth  power  or  the  sum  of  2,  3  ..  or  19  fourth  powers;  this  has 
never  yet  been  fully  demonstrated.  Also  without  proof  is  given  the 
theorem  that  every  even  integer  is  the  sum  of  two  primes  and  every 

1  See  F.  Cajori  in  Bihliotheca  mathematica,  3rd  S.,  Vol.  11,  1911,  pp.  132-137. 

2  For  references  to  the  publications  of  these  writers,  see  F.  Cajori  in  Colorado 
College  Publication,  General  Series  No.  51,  1910,  pp.  186,  187. 


EULER,  LAGRANGE  AXD  LAPLACE  249 

oad  integer  is  a  prime  or  the  sum  of  three  primes.  The  part  relating 
to  even  integers  is  generally  known  as  "Goldbach's  theorem,"  but 
was  first  published  by  Waring.  Christian  Goldbach  communicated 
the  theorem  to  L.  Euler  in  a  letter  of  June  30,  1742,  but  the  letter 
was  not  published  until  1843  (Con.  math.,  P.  H.  Fuss). 

Waring  held  advanced  views  on  the  convergence  of  series.^     He 

taught  that   i+- — I 1 — ~+.  .  .  converges  when  n>i   and  diverges 

wOien  w<i.  He  gave  the  well-known  test  for  convergence  and 
divergence  which  is  often  ascribed  to  A.  L.  Cauchy,  in  which  the 
limit  of  the  ratio  of  the  («-|-i)"'  to  the  n^'^  term  is  considered.  As 
early  as  1757  he  had  found  the  necessary  and  sufficient  relations  which 
must  exist  between  the  coefficients  of  a  quartic  and  quintic  equation, 
for  two  and  for  four  imaginary  roots.  These  criteria  were  obtained 
by  a  new  transformation,  namely  the  one  which  yields  an  equation 
whose  roots  are  the  squares  of  the  differences  of  the  roots  of  the  given 
equation.  To  solve  the  important  problem  of  the  separation  of  the 
roots  Waring  transforms  a  numerical  equation  into  one  whose  roots 
are  reciprocals  of  the  differences  of  the  roots  of  the  given  equation. 
The  reciprocal  of  the  largest  of  the  roots  of  the  transformed  equation 
is  less  than  the  smallest  difference  D,  between  any  two  roots  of  the 
given  equation.  If  M  is  an  upper  limit  of  the  roots  of  the  given  equa- 
tion, then  the  subtraction  of  D,  2D,  7,D,  etc.,  from  ]\f  will  give  values 
which  separate  all  the  real  roots.  In  the  Meditationes  algcbraiccB  of 
1770,  Waring  gives  for  the  first  time  a  process  for  the  approximation 
to  the  values  of  imaginary  roots.  If  x  is  approximately  a+ib,  sub- 
stitute x=a-\-a'-\-{b-\-b')i,  expand  and  reject  higher  powers  of  a'  and 
b' .  Equating  real  numbers  to  each  other  and  imaginary  numbers  to 
each  other,  two  equations  are  obtained  which  yield  values  of  a'  and  b'. 
Etienne  Bezout  (i  730-1 783)  was  a  French  writer  of  popular  mathe- 
matical school-books.  In  his  Thcorie  gcncrale  des  Equations  Alge- 
briques,  1779,  he  gave  the  method  of  elimination  by  linear  equations 
(invented  also  by  L.  Euler).  This  method  was  first  published  by  him 
in  a  memoir  of  1764,  in  which  he  uses  determinants,  without,  however, 
entering  upon  their  theory.  A  beautiful  theorem  as  to  the  degree  of 
the  resultant  goes  by  his  name.  He  and  L.  Euler  both  gave  the  degree 
as  in  general  tn  .  n,  the  product  of  the  orders  of  the  intersecting  loci, 
and  both  proved  the  theorem  by  reducing  the  problem  to  one  of 
elimination  from  an  auxiliary  set  of  linear  equations.  The  determi- 
nant resulting  from  Bezout's  method  is  what  J.  J.  Sylvester  and  later 
writers  call  the  Bezoutiant.  Bezout  fixed  the  degree  of  the  eliminant 
also  for  a  large  number  of  particular  cases.  "One  may  say  that  he 
determined  the  number  of  finite  intersections  of  algebraic  loci,  not 
only  when  all  the  intersections  are  finite,  but  also  when  singular 

1  M.  Cantor,  op.  cit.,  Vol.  IV,  1908,  p.  275. 


25©  A  HISTORY  OF  MATHEMATICS 

points,  or  singular  lines,  planes,  etc.,  at  infinity  occasion  the  with- 
drawal to  infinity  of  certain  of  the  intersection  points;  and  this  at  a 
time  when  the  nature  of  such  singularities  had  not  been  developed."  ^ 
Louis  Arbogaste  (1759-1803)  of  Alsace  was  professor  of  mathe- 
matics at  Strasburg.  His  chief  work,  the  Calcid  des  Derivations,  1800, 
gives  the  method  known  by  his  name,  by  which  the  successive  coeffi- 
cients of  a  development  are  derived  from  one  another  when  the  ex- 
pression is  complicated.  A.  De  Morgan  has  pointed  out  that  the 
true  nature  of  derivation  is  differentiation  accompanied  by  integration. 
In  this  book  for  the  first  time  are  the  symbols  of  operation  separated 

from  those  of  quantity.    The  notation  D^y  for  -3^  is  due  to  him. 

Maria  Gaetana  Agnesi  (17 18-1799)  of  Milan,  distinguished  as  a 
linguist,  mathematician,  and  philosopher,  filled  the  mathematical 
chair  at  the  University  of  Bologna  during  her  father's  sickness. 
Agnesi  was  a  somnambulist.  Several  times  it  happened  to  her  that 
she  went  to  her  study,  while  in  the  somnambuHst  state,  made  a  light, 
and  solved  some  problem  she  had  left  incomplete  when  awake.  In 
the  morning  she  was  surprised  to  find  the  solution  carefully  worked 
out  on  paper.^  In  1748  she  published  her  Instituzioni  Analitiche, 
which  was  translated  into  English  in  1801.  The  "witch  of  Agnesi" 
or  "Versiera"  is  a  cubic  curve  x'^y=a"{a—y)  treated  in  Agnesi's  In- 
stituzioni, but  given  earher  by  P.  Fermat  in  the  form  (a^~x~)  y=a^. 
The  curve  was  discussed  by  Guido  Grandi  in  his  Quadratura  circuit 
et  hyperbolcB,  Pisa,  1703  and  1710.^  In  two  letters  from  Grandi  to 
Leibniz,  in  17 13,  curves  resembling  flowers  are  discussed;  in  1728 
Grandi  published  at  Florence  his  Flares  geometrici.  He  considered 
curves  in  a  plane,  of  the  typep  =  r  sinwco,  and  also  curves  on  a  sphere. 
Recent  studies  along  this  fine  are  due  to  Bodo  Habenicht  (1895), 
E.  W.  Hyde  (1875),  H.  Wieleitner  (1906). 

The  leading  eighteenth  century  historian  of  mathematics  was  Jean 
Etienne  Montucla  (17 25-1 799)  who  pubhshed  a  Histoire  des  mathe- 
matiques,  m  two  volumes,  Paris,  1758.  A  second  edition  of  these  two 
volumes  appeared  in  1799.  A  third  volume,  written  by  Montucla, 
was  partly  printed  when  he  died ;  the  rest  of  it  was  seen  through  the 
press  by  the  astronomer  Joseph  Jerome  le  Francois  de  Lalande 
(173  2-1807),  who  prepared  a  fourth  volume,  mainly  on  the  history  of 
astronomy.^ 

Joseph  Louis  Lagrange  (i 736-1813),  one  of  the  greatest  mathe- 
maticians of  all  times,  was  born  at  Turin  and  died  at  Paris.  He  was 
of  French  extraction.    His  father,  who  had  charge  of  the  Sardinian 

1  H.  S.  White  in  Bull.  Am.  Math.  Soc,  Vol.  15,  1909,  p.  331. 
^  Vlnlermediaire  des  mathematiciens,  Vol.  22,  1915,  p.  241. 
^  G.  Loria,  Ebcne  Ciirven  (F.  Schiitte),  I,  1910,  p.  79. 

■*  For  details  on  other  mathematical  historians,  see  S.  Giinther's  chapter  in 
Cantor,  op.  cit.,  Vol.  IV,  1908,  pp.  1-36. 


EULER,  LAGRANGE  AND  LAPLACE  251 

military  chest,  was  once  wealthy,  but  lost  all  he  had  in  speculation. 
Lagrange  considered  this  loss  his  good  fortune,  for  otherwise  he  might 
not  have  made  mathematics  the  pursuit  of  his  life.  While  at  the 
college  in  Turin  his  genius  did  not  at  once  take  its  true  bent.  Cicero 
and  Virgil  at  first  attracted  him  more  than  Archimedes  and  Newton. 
He  soon  came  to  admire  the  geometry  of  the  ancients,  but  the  perusal 
of  a  tract  of  E.  Halley  roused  his  enthusiasm  for  the  analytical  method, 
in  the  development  of  which  he  was  destined  to  reap  undying  glory. 
He  now  applied  himself  to  mathematics,  and  in  his  seventeenth  year 
he  became  professor  of  mathematics  in  the  royal  miUtary  academy  at 
Turin.  Without  assistance  or  guidance  he  entered  upon  a  course  of 
study  which  in  two  years  placed  him  on  a  level  with  the  greatest  of 
his  contemporaries.  With  aid  of  his  pupils  he  estabUshed  a  society 
which  subsequently  developed  into  the  Turin  Academy.  In  the  first 
five  volumes  of  its  transactions  appear  most  of  his  earlier  papers. 
At  the  age  of  nineteen  he  communicated  to  L.  Euler  a  general  method 
of  deahng  with  "  isoperimetrical  problems,"  known  now  as  the  Cal- 
culus of  Variations.  This  commanded  Euler's  lively  admiration,  and 
he  courteously  withheld  for  a  time  from  publication  some  researches 
of  his  own  on  this  subject,  so  that  the  youthful  Lagrange  might  com- 
plete his  investigations  and  claim  the  invention.  Lagrange  did  quite 
as  much  as  Euler  towards  the  creation  of  the  Calculus  of  Variations. 
As  it  came  from  Euler  it  lacked  an  analytic  foundation,  and  this 
Lagrange  supplied.  He  separated  the  principles  of  this  calculus  from 
geometric  considerations  by  which  his  predecessor  had  derived  them. 
Euler  had  assumed  as  fixed  the  limits  of  the  integral,  i.  e.  the  extrem- 
ities of  the  curve  to  be  determined,  but  Lagrange  removed  this  re- 
striction and  allowed  all  co-ordinates  of  the  curve  to  vary  at  the  same 
time.  Euler  introduced  in  1766  the  name  "calculus  of  variations," 
and  did  much  to  improve  this  science  along  the  lines  marked  out  by 
Lagrange.  Lagrange's  investigations  on  the  calculus  of  variations 
were  published  in  1762,  1771,  1788,  1797,  1806. 

Another  subject  engaging  the  attention  of  Lagrange  at  Turin  was 
the  propagation  of  sound.  In  his  papers  on  this  subject  in  the  Mis- 
cdlanea  Taurinensia,  the  young  mathematician  appears  as  the  critic 
of  I.  Newton,  and  the  arbiter  between  Euler  and  D'Alembert.  By 
considering  only  the  particles  which  are  in  a  straight  line,  he  reduced 
the  problem  to  the  same  partial  differential  equation  that  represents 
the  motions  of  vibrating  strings. 

Vibrating  strings  had  been  discussed  by  Brook  Taylor,  Johann 
Bernoulli  and  his  son  Daniel,  by  D'Alembert  and  L.  Euler.  In  solving 
the  partial  differential  equations,  D'Alembert  restricted  himself  to 
functions  which  can  be  expanded  by  Taylor's  series,  while  Euler 
thought  that  no  restriction  was  necessary,  that  they  could  be  arbi- 
trary, discontinuous.  The  problem  was  taken  up  with  great  skill  by 
Lagrange  who  introduced  new  points  of  view,  but  decided  in  favor  of 


252  A  HISTORY  OF  MATHEMATICS 

Euler.  Later,  de  Condorcet  and  P.  S.  Laplace  stood  on  the  side  of 
D'Alembert  since  in  their  judgment  some  restriction  upon  the  arbi- 
trary functions  was  necessary.  From  the  modern  point  of  view, 
neither  D'Alembert  nor  Euler  was  wholly  in  the  right:  D'Alembert 
insisted  upon  the  needless  restriction  to  functions  with  a  limitless 
number  of  derivatives,  while  Euler  assumed  that  the  ditTerential  and 
integral  calculus  could  be  apphed  to  any  arbitrary  function.^ 

It  now  appears  that  Daniel  BernouUi's  claim  that  his  solution  was 
a  general  one  (a  claim  disputed  by  D'Alembert,  J.  Lagrange  and  L. 
Euler)  was  fully  justified.  The  problem  of  vibrating  strings  stimu- 
lated the  growth  of  the  theory  of  expansions  according  to  trigonometric 
functions  of  multiples  of  the  argument.  H.  Burkhardt  has  pointed 
out  that  there  was  also  another  Une  of  growth  of  this  subject,  namely 
the  growth  in  connection  with  the  problem  of  perturbations,  where 
L.  Euler  started  out  with  the  development  of  the  reciprocal  distance 
of  two  planets  according  to  the  cosine  of  multiples  of  the  angle  be- 
tween their  radii  vectoris. 

By  constant  application  during  nine  years,  Lagrange,  at  the  age 
of  twenty-six,  stood  at  the  summit  of  European  fame.  But  his  intense 
studies  had  seriously  weakened  a  constitution  never  robust,  and  though 
his  physicians  induced  him  to  take  rest  and  exercise,  his  nervous 
system  never  fully  recovered  its  tone,  and  he  was  thenceforth  subject 
to  fits  of  melancholy. 

In  1764  the  French  Academy  proposed  as  the  subject  of  a  prize 
the  theory  of  the  libration  of  the  moon.  It  demanded  an  explanation, 
on  the  principle  of  universal  gravitation,  why  the  moon  always  turns, 
with  but  slight  variations,  the  same  phase  to  the  earth.  Lagrange 
secured  the  prize.  This  success  encouraged  the  Academy  to  propose 
for  a  prize  the  theory  of  the  four  satellites  of  Jupiter, — a  problem  of 
six  bodies,  more  difficult  than  the  one  of  three  bodies  previously 
treated  by  A.  C.  Clairaut,  D'Alembert,  and  L.  Euler.  Lagrange  over- 
came the  difficulties  by  methods  of  approximation.  Twenty-four 
years  afterwards  this  subject  was  carried  further  by  P.  S.  Laplace. 
Later  astronomical  investigations  of  Lagrange  are  on  cometary  per- 
turbations (1778  and  1783),  and  on  Kepler's  problem.  His  researches 
on  the  problem  of  three  bodies  has  been  referred  to  previously. 

Being  anxious  to  make  the  personal  acquaintance  of  leading  mathe- 
maticians, Lagrange  visited  Paris,  where  he  enjoyed  the  stimulating 
delight  of  conversing  with  A.  C.  Clairaut,  D'Alembert,  de  Condorcet, 
the  Abbe  Marie,  and  others.  He  had  planned  a  visit  to  London,  but 
he  fell  dangerously  ill  after  a  dinner  in  Paris,  and  was  compelled  to 
return  to  Turin.  In  1766  L.  Euler  left  Berlin  for  St.  Petersburg,  and 
he  pointed  out  Lagrange  as  the  only  man  capable  of  filling  the  place. 

^  For  details  see  H.  Burkhardt's  Entwicklungen  nach  oscillirenden  Funktionen 
und  Integration  der  Dijfferenlialgleichungen  dcr  mathematlschcn  Physik.  Leipzig, 
1908,  p.  18.    This  is  an  exhaustive  and  valuable  history  of  this  topic. 


EULER,  LAGRANGE  AND  LAPLACE  25^^ 

D'Alembert  recommended  him  at  the  same  time.  Frederick  the  Great 
thereupon  sent  a  message  to  Turin,  expressing  the  wish  of  "the  great- 
est king  of  Europe"  to  have  "the  greatest  mathematician"  at  his 
court.  Lagrange  went  to  BerUn,  and  staid  there  twenty  years.  Find- 
ing all  his  colleagues  married,  and  being  assured  by  their  wives  that 
the  marital  state  alone  is  happy,  he  married.  The  union  was  not  a 
happy  one.  His  wife  soon  died.  Frederick  the  Great  held  him  in 
high  esteem,  and  frequently  conversed  with  him  on  the  advantages 
of  perfect  regularity  of  life.  This  led  Lagrange  to  cultivate  regular 
habits.  He  worked  no  longer  each  day  than  experience  taught  him 
he  could  without  breaking  down.  His  papers  were  carefully  thought 
out  before  he  began  writing,  and  when  he  wrote  he  did  so  without  a 
single  correction. 

During  the  twenty  years  in  Berlin  he  crowded  the  transactions  of 
the  Berlin  Academy  with  memoirs,  and  wrote  also  the  epoch-making 
work  called  the  Mccanique  Analytiqiie.  He  enriched  algebra  by  re- 
searches on  the  solution  of  equations.  There  are  two  methods  of 
solving  directly  algebraic  equations, — that  of  substitution  and  that 
of  combination.  The  former  method  was  developed  by  L.  Ferrari, 
F.  Vieta,  E.  W.  Tchirnhausen,  L.  Euler,  E.  Bezout,  and  Lagrange; 
the  latter  by  C.  A.  Vandermonde  and  Lagrange.^  In  the  method  of 
substitution  the  original  forms  are  so  transformed  that  the  determina- 
tion of  the  roots  is  made  to  depend  upon  simpler  functions  (resolvents). 
In  the  method  of  combination  auxiliary  quantities  are  substituted 
for  certain  simple  combinations  ("types")  of  the  unknown  roots  of 
the  equation,  and  auxiliary  equations  (resolvents)  are  obtained  for 
these  quantities  with  aid  of  the  coefficients  of  the  given  equation.  In 
his  Reflexions  siir  la  resolution  algebriquc  des  equations,  published  in 
Memoirs  of  the  Berlin  Academy  for  the  years  1770  and  1771,  Lagrange 
traced  all  known  algebraic  solutions  of  equations  to  the  uniform  prin- 
ciple consisting  in  the  formation  and  solution  of  equations  of  lower 
degree  whose  roots  are  linear  functions  of  the  required  roots,  and  of 
the  roots  of  unity.  He  showed  that  the  quintic  cannot  be  leduced  in 
this  way,  its  resolvent  being  of  the  sbcth  degree.  In  this  connection 
Lagrange  had  occasion  to  consider  the  number  of  values  a  rational 
function  can  assume  when  its  variables  are  permuted  in  every  possible 
way.  In  these  studies  we  see  the  beginnings  of  the  theory  of  groups. 
The  theorem,  that  the  order  of  a  subgroup  is  a  divisor  of  the  order 
of  the  group  is  practically  established,  and  is  known  now  as  "La- 
grange's theorem,"  although  its  complete  proof  was  iirst  given  about 
thirty  years  later  by  Pietro  Ahbati  (1768-1842)  of  Modena  in  Italy. 
Lagrange's  researches  on  the  theory  of  equations  were  continued  after 
he  left  Berlin.  In  the  Resolution  des  equations  numeriques  (1798)  he 
gave  among  other  things,  a  proof  that  every  equation  must  have  a 
root,— a  theorem  which  before  this  usually  had  been  considered 
1  L.  Matthiessen,  op.  cit.,  pp.  80-84. 


254  A  HISTORY  OF  MATHEMATICS 

self-evident.  Other  proofs  of  this  were  given  by  J.  R.  Argand,  K.  F. 
Gauss,  and  A.  L.  Cauchy.  In  a  note  to  the  above  work  Lagrange  uses 
Fermat's  theorem  and  certain  suggestions  of  Gauss  in  effecting  a  com- 
plete algebraic  solution  of  any  binomial  equation. 

In  the  Berhn  Memoires  for  the  year  1767  Lagrange  contributed  a 
paper,  Sur  la  resolution  des  equations  numeriques.  He  explains  the 
separation  of  the  real  roots  by  substituting  for  x  the  terms  of  the 
progression,  o,  D,  2D,  .  .  .,  where  D  must  be  less  than  the  least  dif- 
ference between  the  roots.  Lagrange  suggested  three  ways  of  com- 
puting D:  One  way  in  1767,  another  in  1795  and  a  third  in  1798.  The 
first  depends  upon  the  equation  of  the  squared  differences  of  the  roots 
of  the  given  equation.  E.  Waring  before  this  had  derived  this  im- 
portant equation,  but  in  1767  Lagrange  had  not  yet  seen  Waring's 
writings.  Lagrange  finds  equal  roots  by  computing  the  highest  com- 
mon factor  between /(x)  and/'(.r).  He  proceeds  to  develop  a  new 
mode  of  approximation,  that  by  continued  fractions.  P.  A.  Cataldi 
had  used  these  fractions  in  extracting  square  roots.  Lagrange  enters 
upon  greater  details  in  his  Additions  to  his  paper  of  1767.  Unhke 
the  older  methods  of  approximation,  Lagrange's  has  no  cases  of 
failure.  "  Cette  methode  ne  laisse,  ce  me  semble,  rien  a  desirer,"  yet, 
though  theoretically  perfect,  it  yields  the  root  in  the  form  of  a  con- 
tinued fraction  which  is  undesirable  in  practice. 

While  in  Berlin  Lagrange  pubhshed  several  papers  on  the  theory 
of  numbers.  In  1769  he  gave  a  solution  in  integers  of  indeterminate 
equations  of  the  second  degree,  which  resembles  the  Hindu  cycHc 
method;  he  was  the  first  to  prove,  in  1771,  "Wilson's  theorem,"  enun- 
ciated by  an  Englishman,  John  Wilson,  and  first  published  by  E. 
Waring  in  his  Meditationes  Algebraicce;  he  investigated  in  1775  under 
what  conditions  =i=  2  and  =±=  5  (—  i  and  db  3  having  been  discussed  by 
L.  Euler)  are  quadratic  residues,  or  non-residues  of  odd  prime  num- 
bers, q;  he  proved  in  1770  Bachet  de  Meziriac's  theorem  that  every 
integer  is  equal  to  the  sum  of  four,  or  a  less  number,  of  squares.  He 
pioved  Fermat's  theorem  on  x"'+y"-  =  z'^,  for  the  case  w=4,  also  Fer- 
mat's theorem  that,  if  a^+P=c^,  then  ab  is  not  a  square. 

In  his  memoir  on  Pyramids,  1773,  Lagrange  made  considerable  use 
of  determinants  of  the  third  order,  and  demonstrated  that  the  square 
of  a  determinant  is  itself  a  determinant.  He  never,  however,  dealt 
explicitly  and  directly  with  determinants;  he  simply  obtained  acci- 
dentally identities  which  are  now  recognized  as  relations  between 
determinants. 

Lagrange  wrote  much  on  differential  equations.  Though  the  sub- 
ject of  contemplation  by  the  greatest  mathematicians  (L.  Euler, 
D'Alembert,  A.  C.  Clairaut,  J.  Lagrange,  P.  S.  Laplace),  yet  more 
than  other  branches  of  mathematics  do  they  resist  the  systematic 
apphcation  of  fixed  methods  and  principles.  The  subject  of  singular 
solutions,  which  had  been  taken  up  by  P.  S.  Laplace  in  177 1  and  1774, 


EULER,  LAGRANGE  AND  LAPLACE  255 

was  investigated  by  Lagrange  who  gave  the  derivation  of  a  singular 
solution  from  the  general  solution  as  well  as  from  the  differential 
equation  itself.  Lagrange  brought  to  view  the  relation  of  singular 
solutions  to  envelopes.  Nevertheless,  he  failed  to  remove  all  mystery 
surrounding  this  subtle  subject.  An  inconsistency  in  his  theorems 
caused  about  1870  a  complete  reconsideration  of  the  entire  theory  of 
singular  solutions.  Lagrange's  treatment  is  given  in  his  Calcul  des 
Fonctions,  Lessons  14-17.  He  generalized  Euler's  researches  on  total 
dilTerential  equations  of  two  variables,  and  of  the  ninth  order;  he 
gave  a  solution  of  partial  differential  equations  of  the  first  order  {Berlin 
Memoirs,  1772  and  1774),  and  spoke  of  their  singular  solutions,  ex- 
tending their  solution  in  Memoirs  of  1779  and  17S5  to  equations  of 
any  number  of  variables.  The  Memoirs  of  1772  and  1774  were  refined 
in  certain  points  by  a  young  mathematician  Paul  Charpit  (?-i784) 
whose  method  of  solution  was  first  printed  in  LacroLx's  Traite  du 
cakul,  2.  Ed.,  Paris,  1814,  T.  II,  p.  548.  The  discussion  on  partial 
differential  equations  of  the  second  order,  carried  on  by  D'Alembert, 
Euler,  and  Lagrange,  has  already  been  referred  to  in  our  account  of 
D'Alembert. 

While  in  Berlin,  Lagrange  wrote  the  " Mecanique  Analylique,''  the 
greatest  of  his  works  (Paris,  178S).  From  the  principle  of  virtual 
velocities  he  deduced,  with  aid  of  the  calculus  of  variations,  the  whole 
system  of  mechanics  so  elegantly  and  harmoniously  that  it  may  fitly 
be  called,  in  Sir  William  Rowan  Hamilton's  words,  "a  kind  of  scien- 
tific poem."  It  is  a  most  consummate  example  of  analytic  generality. 
Geometrical  figures  are  nowhere  allowed.  "On  ne  trouvera  point  de 
figures  dans  cet  ouvrage"  (Preface).  The  two  divisions  of  mechanics 
— statics  and  dynamics — are  in  the  first  four  sections  of  each  carried 
out  analogously,  and  each  is  prefaced  by  a  historic  sketch  of  principles. 
Lagrange  formulated  the  principle  of  least  action.  In  their  original 
form,  the  equations  of  motion  involve  the  co-ordinates  x,  y,  z,  of  the 
different  particles  m  or  d?n  of  the  system.  But  x,  y,  z,  are  in  general 
not  independent,  and  Lagrange  introduced  in  place  of  them  any 
variables  i,  \p,  4>,  whatever,  determining  the  position  of  the  point  at 
the  time.  These  "generalized  co-ordinates"  may  be  taken  to  be  inde- 
pendent.   The  equations  of  motion  may  now  assume  the  form 

d_dT_dT 
dtd$'    d$'^      °' 

or  when  H,  i/',  <^,  .  .  .  are  the  partial  differential  coefficients  with 
respect  to  $,il/,  <f>,  .  .  .  of  one  and  the  same  function  V,  then  the  form 

ddJ^_dT    dV_ 
dtd$'    d^^d$~°- 

The  latter  is  par  excellence  the  Lagrangian  form  of  the  equations  of 
inotion.     With  Lagrange  originated  the  remark  that  mechanics  may 


256  A  HISTORY  OF  MATHEMATICS 

be  regarded  as  a  geometry  of  four  dimensions.  To  him  falls  the  honor 
of  the  introduction  of  the  potential  into  dynamics.  Lagrange  was 
anxious  to  have  his  Mecanique  Analytique  published  in  Paris.  The 
work  was  ready  for  print  in  1786,  but  not  till  17S8  could  he  find  a 
publisher,  and  then  only  with  the  condition  that  after  a  few  years 
he  would  purchase  all  the  unsold  copies.  The  work  was  edited  by 
A.  M.  Legendre. 

After  the  death  of  Frederick  the  Great,  men  of  science  were  no 
longer  respected  in  Germany,  and  Lagrange  accepted  an  invitation 
of  Louis  XVI  to  migrate  to  Paris.  The  French  queen  treated  him 
with  regard,  and  lodging  was  procured  for  him  in  the  Louvre.  But 
he  was  seized  with  a  long  attack  of  melancholy  which  destroyed  his 
taste  for  mathematics.  For  two  years  his  printed  copy  of  the  Me- 
canique, fresh  from  the  press, — the  work  of  a  quarter  of  a  century, — ■■ 
lay  unopened  on  his  desk.  Through  A.  L.  Lavoisier  he  became  in- 
terested in  chemistry,  which  he  found  "as  easy  as  algebra."  The 
disastrous  crisis  of  the  French  Revolution  aroused  him  again  to  ac- 
tivity. About  this  time  the  young  and  accomplished  daughter  of  the 
astronomer  P.  C.  Lemonnier  took  compassion  on  the  sad,  lonely 
Lagrange,  and  insisted  upon  marrying  him.  Her  devotion  to  him 
constituted  the  one  tie  to  life  which  at  the  approach  of  death  he  found 
it  hard  to  break. 

He  was  made  one  of  the  commissioners  to  estabUsh  weights  and 
measures  having  units  founded  on  nature.  Lagrange  strongly  favored 
the  decimal  subdivision.  Such  was  the  moderation  of  Lagrange's 
character,  and  such  the  universal  respect  for  him,  that  he  was  retained 
as  president  of  the  commission  on  weights  and  measures  even  after 
it  had  been  purified  by  the  Jacobins  by  striking  out  the  names  of  A.  L. 
Lavoisier,  P.  S.  Laplace,  and  others.  Lagrange  took  alarm  at  the 
fate  of  Lavoisier,  and  planned  to  return  to  Berlin,  but  at  the  estab- 
lishment of  the  Ecole  Nonnale  in  1795  i^i  Paris,  he  was  induced  to 
accept  a  professorship.  Scarcely  had  he  time  to  elucidate  the  founda- 
tions of  arithmetic  and  algebra  to  young  pupils,  when  the  school  was 
closed.  His  additions  to  the  algebra  of  L.  Euler  were  prepared  at 
this  time.  In  1 797  the  Ecole  Polytechnique  was  founded,  with  Lagrange 
as  one  of  the  professors.  The  earliest  triumph  of  this  institution  was 
the  restoration  of  Lagrange  to  analysis.  His  mathematical  activity 
burst  out  anew.  He  brought  forth  the  Theorie  des  fonctions  analyiiques 
(1797),  Lemons  sur  le  calcul  des  fonctions,  a  treatise  on  the  same  lines 
as  the  preceding  (1801),  and  the  Resolution  des  equations  numeriques 
(1798),  which  includes  papers  pubhshed  much  earlier;  his  memoir, 
Nouvelle  methode  pour  rcsoudre  les  equations  litterales  par  le  moyen  des 

series,  pubhshed  1770,  gives  the  notation  «/''  for  -^,  which  occurs 

however  much  earlier  in  a  part  of  a  memoir  by  Francois  Daviet  de 


EULER,  LAGRANGE  AND  LAPLACE  257 

Foncenex  in  the  Miscellanea  Tatirinensia  for  1759,  believed  to  have 
been  written  for  Foncenex  by  Lagrange  himself.'^  In  18 10  he  began 
a  thorough  revision  of  his  Mccanique  analytiqiic,  but  he  died  before 
its  completion. 

The  Thcorie  dcs  fondions,  the  germ  of  which  is  found  in  a  memoir 
of  his  of  1772,  aimed  to  place  the  principles  of  the  calculus  upon  a 
sound  foundation  by  relieving  the  mind  of  the  difficult  conception  of 
a  limit.  John  Landen's  residual  calculus,  professing  a  similar  object, 
was  unknown  to  him.  In  a  letter  to  L.  Euler  of  Nov.  24,  1759,  La- 
grange says  that  he  believed  he  had  developed  the  true  metaphysics 
of  the  calculus;  at  that  time  he  seems  to  have  been  convinced  that 
the  use  of  infinitesimals  was  rigorous.  He  "  used  both  the  infinitesimal 
method  and  the  method  of  derived  functions  side  by  side  during  his 
whole  life"  (Jourdain).  Lagrange  attempted  to  prove  Taylor's 
theorem  (the  power  of  which  he  was  the  first  to  point  out)  by  simple 
algebra,  and  then  to  develop  the  entire  calculus  from  that  theorem. 
The  principles  of  the  calculus  were  in  his  day  involved  in  philosophic 
difficulties  of  a  serious  nature.  The  infinitesimals  of  G.  W.  LeilDniz 
had  no  satisfactory  metaphysical  basis.  In  the  differential  calculus 
of  L.  Euler  they  were  treated  as  absolute  zeros.  In  I.  Newton's  limit- 
ing ratio,  the  magnitudes  of  which  it  is  the  ratio  cannot  be  found, 
for  at  the  moment  when  they  should  be  caught  and  equated,  there  is 
neither  arc  nor  chord.  The  chord  and  arc  were  not  taken  by  Newton 
as  equal  before  vanishing,  nor  after  vanishing,  but  when  they  vanish. 
"That  method,"  said  Lagrange,  "has  the  great  inconvenience  of  con- 
sidering quantities  in  the  state  in  which  they  cease,  so  to  speak,  to  be 
quantities;  for  though  we  can  always  well  conceive  the  ratios  of  two 
quantities,  as  long  as  they  remain  finite,  that  ratio  offers  to  the  mind 
no  clear  and  precise  idea,  as  soon  as  its  terms  become  both  nothing 
at  the  same  time."  D'Alembert's  method  of  limits  was  much  the 
same  as  the  method  of  prime  and  ultimate  ratios.  When  Lagrange 
endeavored  to  free  the  calculus  of  its  metaphysical  difficulties,  by 
resorting  to  common  algebra,  he  avoided  the  whirlpool  of  Charybdis 
only  to  suffer  wreck  against  the  rocks  of  Scylla.  The  algebra  of  his 
day,  as  handed  down  to  him  by  L.  Euler,  was  founded  on  a  false 
view  of  infinity.  No  rigorous  theory  of  infinite  series  had  then  been 
established.  Lagrange  proposed  to  define  the  differential  coefficient 
of  f{x)  with  respect  to  x  as  the  coefficient  of  h  in  the  expansion  of 
f(x+h)  by  Taylor's  theorem,  and  thus  to  avoid  all  reference  to  limits. 
But  he  used  infinite  series  without  ascertaining  carefully  that  they 
were  convergent,  and  his  proof  that  /(x-j-A)  can  always  be  expanded 
in  a  series  of  ascending  powers  of  //,  labors  under  serious  defects. 
Though  Lagrange's  method  of  developing  the  calculus  was  at  first 
greatly  applauded,  its  defects  were  fatal,  and  to-day  his  "method  of 

'  Philip  E.  B.  Jourdain  In  Proceed,  jlh  Intern.  Congress,  Cambridge,  IQI2,  Cam- 
bridge, 19 13,  Vol.  II,  p.  540. 


258  A  HISTORY  OF  MATHEMATICS 

derivatives,"  as  it  was  called,  has  been  generally  abandoned.  He 
introduced  a  notation  of  his  own,  but  it  was  inconvenient,  and  was 
abandoned  by  him  in  the  second  edition  of  his  Mecanique,  in  which 
he  used  infinitesimals.  The  primary  object  of  the  Theorie  des  fonctions 
was  not  attained,  but  its  secondary  results  were  far-reaching.  It 
was  a  purely  abstract  mode  of  regarding  functions,  apart  from  geo- 
metrical or  mechanical  considerations.  In  the  further  development 
of  higher  analysis  a  function  became  the  leading  idea,  and  Lagrange's 
work  may  be  regarded  as  the  starting-point  of  the  theory  of  functions 
as  developed  by  A.  L.  Cauchy,  G.  F.  B.  Riemann,  K.  Weierstrass, 
and  others. 

The  first  to  doubt  the  rigor  of  Lagrange's  exposition  of  the  calculus 
were  Abel  Biirja  (i752-i8i6)of  Berhn,  the  two  PoHsh  mathematicians 
//.  Wronski  and  /.  B.  Sniadecki  (i 756-1830),  and  the  Bohemian 
B.  Bolzano,  who  were  all  men  of  limited  acquaintance  and  influence. 
It  remained  for  A.  L.  Cauchy  really  to  initiate  the  period  of  greater 
rigor. 

Instructive  is  C.  E.  Picard's  characterization  of  the  time  of  La- 
grange: "In  all  this  period,  especially  in  the  second  half  of  the  eight- 
eenth century,  what  strikes  us  with  admiration  and  is  also  somewhat 
confusing,  is  the  extreme  importance  of  the  applications  realized, 
while  the  pure  theory  appeared  still  so  ill  assured.  One  perceives  it 
when  certain  questions  are  raised  like  the  degree  of  arbitrariness  in 
the  integral  of  vibrating  chords,  which  gives  place  to  an  interminable 
and  inconclusive  discussion.  Lagrange  appreciated  these  insufficiencies 
when  he  published  his  theory  of  analytic  functions,  where  he  strove 
to  give  a  precise  foundation  to  analysis.  One  cannot  too  much 
admire  the  marvellous  presentiment  he  had  of  the  role  which  the 
functions,  which  with  him  we  call  analytic,  were  to  play;  but  we  may 
confess  that  we  stand  astonished  before  the  demonstration  he  be- 
lieved to  have  given  of  the  possibility  of  the  development  of  a  function 
in  Taylor's  series."  ^ 

In  the  treatment  of  infinite  series  Lagrange  displayed  in  his  earher 
writings  that  laxity  common  to  all  mathematicians  of  his  time,  ex- 
cepting Nicolaus  Bernoulli  II  and  D'AIembert,  But  his  later  articles 
mark  the  beginning  of  a  period  of  greater  rigor.  Thus,  in  the  Calcul  des 
fonctions  he  gives  his  theorem  on  the  limits  of  Taylor's  theorem.  La- 
grange's mathematical  researches  extended  to  subjects  which  have 
not  been  mentioned  here — such  as  probabilities,  finite  differences, 
ascending  continued  fractions,  elliptic  integrals.  Everywhere  his 
wonderful  powers  of  generalization  and  abstraction  are  made  manifest. 
In  that  respect  he  stood  without  a  peer,  but  his  great  contemporary, 
P.  S.  Laplace,  surpassed  him  in  practical  sagacity.  Lagrange  was 
content  to  leave  the  application  of  his  general  results  to  others,  and 
some  of  the  most  important  researches  of  Laplace  (particularly  those 
'  Con  '-CSS  cf  Arls  and  Science,  St.  Louis,  1904,  Vol.  1,  p.  503. 


EULER,  LAGRANGE  AND  LAPLACE  259 

on  the  velocity  of  sound  and  on  the  secular  acceleration  of  the  moon) 
are  implicitly  contained  in  Lagrange's  works. 

Lagrange  was  an  extremely  modest  man,  eager  to  avoid  contro- 
versy, and  even  timid  in  conversation.  He  spoke  in  tones  of  doubt, 
and  his  first  words  generally  were,  "  Je  ne  sais  pas."  He  would  never 
allow  his  portrait  to  be  taken,  and  the  only  ones  that  were  secured 
were  sketched  without  his  knowledge  by  persons  attending  the  meet- 
ings of  the  Institute. 

Pierre  Simon  Laplace  (i 749-1827)  was  born  at  Beaumont-en-Auge 
in  Normandy.  Very  little  is  known  of  his  early  life.  When  at  the 
height  of  his  fame  he  was  loath  to  speak  of  his  boyhood,  spent  in 
poverty.  His  father  was  a  small  farmer.  Some  rich  neighbors  who 
recognized  the  boy's  talent  assisted  him  in  securing  an  education. 
As  an  extern  he  attended  the  military  school  in  Beaumont,  where  at 
an  early  age  he  became  teacher  of  mathematics.  At  eighteen  he  went 
to  Paris,  armed  with  letters  of  recommendation  to  D'Alembert,  who 
was  then  at  the  height  of  his  fame.  The  letters  remained  unnoticed, 
but  young  Laplace,  undaunted,  wrote  the  great  geometer  a  letter  on 
the  principles  of  mechanics,  which  brought  the  following  enthusiastic 
response:  "You  needed  no  introduction;  you  have  recommended  your- 
self; my  support  is  your  due."  D'Alembert  secured  him  a  position 
at  the  Ecole  Militaire  of  Paris  as  professor  of  mathematics.  His  future 
was  now  assured,  and  he  entered  upon  those  profound  researches 
which  brought  him  the  title  of  "the  Newton  of  France."  With 
wonderful  mastery  of  analysis,  Laplace  attacked  the  pending  problems 
in  the  application  of  the  law  of  gravitation  to  celestial  motions.  Dur- 
ing the  succeeding  fifteen  years  appeared  most  of  his  original  contri- 
butions to  astronomy.  His  career  was  one  of  almost  uninterrupted 
prosperity.  In  1784  he  succeeded  E.  Bezout  as  examiner  to  the  royal 
artillery,  and  the  following  year  he  became  member  of  the  Academy 
of  Sciences.  He  was  made  president  of  the  Bureau  of  Longitude;  he 
aided  in  the  introduction  of  the  decimal  system,  and  taught,  with 
J.  Lagrange,  mathematics  in  the  Ecole  Normale.  When,  during  the 
Revolution,  there  arose  a  cry  for  the  reform  of  everything,  even  of 
the  calendar,  Laplace  suggested  the  adoption  of  an  era  beginning  with 
the  year  1250,  when,  according  to  his  calculation,  the  major  axis  of 
the  earth's  orbit  had  been  perpendicular  to  the  equinoctial  line.  The 
year  was  to  begin  with  the  vernal  equinox,  and  the  zero  meridian  was 
to  be  located  east  of  Paris  by  185.30  degrees  of  the  centesimal  division 
of  the  quadrant,  for  by  this  meridian  the  beginning  of  his  proposed 
era  fell  at  midnight.  But  the  revolutionists  rejected  this  scheme,  and 
made  the  start  of  the  new  era  coincide  with  the  beginning  of  the 
glorious  French  Republic.^ 

Laplace  was  justly  admired  throughout  Europe  as  a  most  sagacious 

1  Rudolf  Wolf,  Geschichte  der  Astronomic,  Miinchen,  1877,  p.  334. 


26o  A  HISTORY  OF  MATHEMATICS 

and  profound  scientist,  but,  unhappily  for  his  reputation,  he  strove 
not  only  after  greatness  in  science,  but  also  after  political  honors. 
The  poHtical  career  of  this  eminent  scientist  was  stained  by  servility 
and  suppleness.  After  the  i8th  of  Brumaire,  the  day  when  Napoleon 
was  made  emperor,  Laplace's  ardor  for  republican  principles  suddenly 
gave  way  to  a  great  devotion  to  the  emperor.  Napoleon  rewarded 
this  devotion  by  giving  him  the  post  of  minister  of  the  interior,  but 
dismissed  him  after  six  months  for  incapacity.  Said  Napoleon,  "La- 
place ne  saisissait  aucune  question  sous  son  veritable  point  de  vue;  il 
cherchait  des  subtiHtes  partout,  n'avait  que  des  idees  problematiques, 
et  portait  eniin  I'esprit  des  infiniment  petits  jusque  dans  I'administra- 
tion."  Desirous  to  retain  his  allegiance.  Napoleon  elevated  him  to 
the  Senate  and  bestowed  various  other  honors  upon  him.  Neverthe- 
less, he  cheerfully  gave  his  voice  in  1S14  to  the  dethronement  of  his 
patron  and  hastened  to  <^ender  his  services  to  the  Bourbons,  thereby 
earning  the  title  of  marquis.  This  pettiness  of  his  character  is  seen 
in  his  writings.  The  first  edition  of  the  Systeme  du  monde  was  dedi- 
cated to  the  Council  of  Five  Hundred.  To  the  third  volume  of  the 
Mecanique  Celeste  is  prefixed  a  note  that  of  all  the  truths  contained 
in  the  book,  the  one  most  precious  to  the  author  was  the  declaration  he 
thus  made  of  gratitude  and  devotion  to  the  peace-maker  of  Europe. 
After  this  outburst  of  affection,  we  are  surprised  to  find  in  the  editions 
of  the  Thcorie  analytique  des  probabilites,  which  appeared  after  the 
Restoration,  that  the  original  dedication  to  the  emperor  is  suppressed. 

Though  supple  and  servile  in  politics,  it  must  be  said  that  in  religion 
and  science  Laplace  never  misrepresented  or  concealed  his  own  con- 
victions however  distasteful  they  might  be  to  others.  In  mathematics 
and  astronomy  his  genius  shines  with  a  lustre  excelled  by  few.  Three 
great  works  did  he  give  to  the  scientific  world, — the  Mecanique  Celeste, 
the  Exposition  du  systeme  du  monde,  and  the  Thcorie  analytique  des 
probabilites.  Besides  these  he  contributed  important  memoirs  to  the 
French  Academy. 

We  first  pass  in  brief  review  his  astronomical  researches.  In  1773 
he  brought  out  a  paper  in  which  he  proved  that  the  mean  motions 
or  mean  distances  of  planets  are  invariable  or  merely  subject  to  small 
periodic  changes.  This  was  the  first  and  most  important  step  in  his 
attempt  to  establish  the  stability  of  the  solar  system.^  To  I.  Newton 
and  also  to  L.  Euler  it  had  seemed  doubtful  whether  forces  so  numer- 
ous, so  variable  in  position,  so  different  in  intensity,  as  those  in  the 
solar  system,  couid  be  capable  of  maintaining  permanently  a  condition 
of  equilibrium.  Newton  was  of  the  opinion  that  a  powerful  hand 
must  intervene  from  time  to  time  to  repair  the  derangements  occa- 
sioned by  the  mutual  action  of  the  different  bodies.  This  paper  was 
the  beginning  of  a  series  of  profound  researches  by  J.  Lagrange  and 

^  D.  F.  J.  Arago,  "Eulogy  on  Laplace,"  translated  by  B.  Powell,  Smithsonian 
Report,  1874. 


EULER,  EAGRANGE  AND  LAPLACE  261 

Laplace  on  the  limits  of  variation  of  the  various  elements  of  planetary 
orbits,  in  which  the  two  great  mathematicians  alternately  surpassed 
and  supplemented  each  other.  Laplace's  first  paper  really  grew  out 
of  researches  on  the  theory  of  Jupiter  and  Saturn.  The  behavior  of 
these  planets  had  been  studied  by  L.  Euler  and  J.  Lagrange  without 
receiving  satisfactory  explanation.  Observation  revealed  the  ex- 
istence of  a  steady  acceleration  of  the  mean  motions  of  our  moon  and 
of  Jupiter  and  an  equally  strange  diminution  of  the  mean  motion  of 
Saturn.  It  looked  as  though  Saturn  miglit  eventually  leave  the 
planetary  system,  while  Jupiter  would  fall  into  the  sun,  and  the  moon 
upon  the  earth.  Laplace  finally  succeeded  in  showing,  in  a  paper  of 
1784-1786,  that  these  variations  (called  the  "great  inequality")  be- 
longed to  the  class  of  ordinary  periodic  perturbations,  depending  upon 
the  law  of  attraction.  The  cause  of  so  influential  a  perturbation  was 
found  in  the  commensurability  of  the  mean  motion  of  the  two  planets. 

In  the  study  of  the  Jovian  system,  Laplace  was  enabled  to  deter- 
mine the  masses  of  the  moons.  He  also  discovered  certain  very 
remarkable,  simple  relations  between  the  movements  of  those  bodies, 
known  as  "Laws  of  Laplace."  His  theory  of  these  bodies  was  com- 
pleted in  papers  of  178S  and  1789.  These,  as  well  as  the  other  papers 
here  mentioned,  were  published  in  the  Memoirs  prcsentes  par  divers 
savans.  The  year  1787  was  made  memorable  by  Laplace's  announce- 
ment that  the  lunar  acceleration  depended  upon  the  secular  changes 
in  the  eccentricity  of  the  earth's  orbit.  This  removed  all  doubt  then 
existing  as  to  the  stabiUty  of  the  solar  system.  The  universal  validity 
of  the  law  of  gravitation  to  explain  all  motion  in  the  solar  system 
seemed  established.  That  system,  as  then  known,  was  at  last  found 
to  be  a  complete  machine. 

In  1796  Laplace  published  his  Exposition  dii  systcme  du  monde, 
a  non-mathematical  popular  treatise  on  astronomy,  ending  with  a 
sketch  of  the  history  of  the  science.  In  this  work  he  enunciates  for 
the  first  time  his  celebrated  nebular  hypothesis.  A  similar  theory 
had  been  previously  proposed  by  L  Kant  in  1755,  and  by  E.  Sweden- 
borg;  but  Laplace  does  not  appear  to  have  been  aware  of  this. 

Laplace  conceived  the  idea  of  writing  a  work  which  should  contain 
a  complete  analytical  solution  of  the  mechanical  problem  presented  by 
the  solar  system,  without  deriving  from  observation  any  but  indis- 
pensable data.  The  result  was  the  Mecanique  Celeste,  which  is  a 
systematic  presentation  embracing  all  the  discoveries  of  I.  Newton, 
A.  C.  Clairaut,  D'Alembert,  L.  Euler,  J.  Lagrange,  and  of  Laplace 
himself,  on  celestial  mechanics.  The  first  and  second  volumes  of  this 
work  were  published  in  1799;  the  third  appeared  in  1802,  the  fourth 
in  1S05.  Of  the  fifth  volume,  Books  XI  and  XII  were  published  in 
1S23;  Books  XIII,  XIV,  XV  in  1824,  and  Book  XVI  in  1825.  The 
first  two  volumes  contain  the  general  theory  of  the  motions  and  figure 
of  celestial  bodies.    The  third  and  fourth  volumes  give  special  theories 


262  A  HISTORY  OF  MATHEMATICS 

of  celestial  motions, — treating  particularly  of  motions  of  comets,  oi 
our  moon,  and  of  other  satellites.  The  fifth  volume  opens  with  a 
brief  history  of  celestial  mechanics,  and  then  gives  in  appendices  the 
results  of  the  author's  later  researches.  The  Mecanique  Celeste  was 
such  a  master-piece,  and  so  complete,  that  Laplace's  immediate  suc- 
cessors were  able  to  add  comparatively  little.  The  general  part  of 
the  work  was  translated  into  German  by  Johann  Karl  Burkhardt 
(1773-1825),  and  appeared  in  Berlin,  1800-1S02.  Nathaniel  Bowditch 
(1773-1838)  brought  out  an  edition  in  English,  with  an  extensive  com- 
mentary, in  Boston,  1S29-1839.  The  Mccanique  Celeste  is  not  easy 
reading.  The  difficulties  lie,  as  a  rule,  not  so  much  in  the  subject 
itself  as  in  the  want  of  verbal  explanation.  A  complicated  chain  of 
reasoning  receives  often  no  explanation  whatever.  J.  B.  Biot,  who 
assisted  Laplace  in  revising  the  work  for  the  press,  tells  that  he  once 
asked  Laplace  some  explanation  of  a  passage  in  the  book  which  had 
been  written  not  long  before,  and  that  Laplace  spent  an  hour  endeavor- 
ing to  recover  the  reasoning  which  had  been  carelessly  suppressed 
with  the  remark,  "II  est  facile  de  voir."  Notwithstanding  the  impor- 
tant researches  in  the  work,  which  are  due  to  Laplace  himself,  it 
naturally  contains  a  great  deal  that  is  drawn  from  his  predecessors. 
It  is,  in  fact,  the  organized  result  of  a  century  of  patient  toil.  But 
Laplace  frequently  neglects  properly  to  acknowledge  the  source  from 
which  he  draws,  and  lets  the  reader  infer  that  theorems  and  formulae 
due  to  a  predecessor  are  really  his  own. 

We  are  told  that  when  Laplace  presented  Napoleon  with  a  copy 
of  the  Mecanique  Celeste,  the  latter  made  the  remark,  "M.  Laplace, 
they  tell  m.e  you  have  written  this  large  book  on  the  system  of  the 
universe,  and  have  never  even  mentioned  its  Creator."  Laplace  is 
said  to  have  replied  bluntly,  "Je  n'avais  pas  besoin  de  cette  hy- 
pothese-la."  This  assertion,  taken  literally,  is  impious,  but  may  it 
not  have  been  intended  to  convey  a  meaning  somewhat  different 
from  its  literal  one?  I.  Newton  was  not  able  to  explain  by  his  law  of 
gravitation  all  questions  arising  in  the  mechanics  of  the  heavens. 
Thus,  being  unable  to  show  that  the  solar  system  was  stable,  and 
suspecting  in  fact  that  it  was  unstable,  Newton  expressed  the  opinion 
that  the  special  intervention,  from  time  to  time,  of  a  powerful  hand 
was  necessary  to  preserve  order.  Now  Laplace  thought  that  he  had 
proved  by  the  law  of  gravitation  that  the  solar  system  is  stable,  and 
in  that  sense  may  be  said  to  have  felt  no  necessity  for  reference  to  the 
Almighty. 

We  now  proceed  to  researches  which  belong  more  properly  to  pure 
mathematics.  Of  these  the  most  conspicuous  are  on  the  theory  of 
probability.  Laplace  has  done  more  towards  advancing  this  subject 
than  any  one  other  investigator.  He  published  a  series  of  papers, 
the  main  results  of  which  were  collected  in  his  Thcorie  analytique  des 
probabilites,  1S12.    The  third  edition  (1820)  consists  of  an  introduction 


EULER,  LAGRANGE  AND  LAPLACE  263 

and  two  books.  The  introduction  was  published  separately  under 
the  title,  Essai  philosopJiique  sur  les  probabUUcs,  and  is  an  admirable 
and  masterly  exposition  without  the  aid  of  analytical  formulae  of  the 
l)rinciplcs  and  applications  of  the  science.  The  first  book  contains 
the  theory  of  generating  functions,  which  are  applied,  in  the  second 
book,  to  the  theory  of  probability.  Laplace  gives  in  his  work  on 
probability  his  method  of  approximation  to  the  values  of  definite 
integrals. '  The  solution  of  linear  differential  equations  was  reduced 
by  him  to  definite  integrals.  The  use  of  partial  difference  equations 
was  introduced  into  the  study  of  probability  by  him  about  the  same 
time  as  by  J.  Lagrange.  One  of  the  most  important  parts  of  the 
work  is  the  application  of  probability  to  the  method  of  least  squares, 
which  is  shown  to  give  the  most  probable  as  well  as  the  most  conven- 
ient results. 

Laplace's  work  on  probability  is  very  difficult  reading,  particularly 
the  part  on  the  method  of  least  squares.  The  analytical  processes 
are  by  no  means  clearly  estabhshed  or  free  from  error.  "No  one  was 
more  sure  of  giving  the  result  of  analytical  processes  correctly,  and 
no  one  ever  took  so  little  care  to  point  out  the  various  small  con- 
siderations on  which  correctness  depends"  (De  Morgan).  Laplace's 
comprehensive  work  contains  all  of  his  own  researches  and  much 
derived  from  other  writers.  He  gives  masterly  expositions  of  the 
Problem  of  Points,  of  Jakob  Bernoulli's  theorem,  of  the  problems  taken 
from  Bayes  and  Count  de  Buffon.  In  this  work  as  in  his  Mecaniqiie 
Celeste,  Laplace  is  not  in  the  habit  of  giving  due  credit  to  writers  that 
preceded  him.  A.  De  Morgan^  says  of  Laplace:  "There  is  enough 
originating  from  himself  to  make  any  reader  wonder  that  one  who 
could  so  well  afford  to  state  what  he  had  taken  from  others,  should 
have  set  an  example  so  dangerous  to  his  own  claims." 

Of  Laplace's  papers  on  the  attraction  of  ellipsoids,  the  most  im- 
portant is  the  one  published  in  1785,  and  to  a  great  extent  reprinted 
in  the  third  volume  of  the  Mccanique  Celeste.  It  gives  an  exhaustive 
treatment  of  the  general  problem  of  attraction  of  any  ellipsoid  upon 
a  particle  situated  outside  or  upon  its  surface.  Spherical  harmonics, 
or  the  so-called  "Laplace's  coefficients,"  constitute  a  powerful  analytic 
engine  in  the  theory  of  attraction,  in  electricity,  and  magnetism.  The 
theory  of  spherical  harmonics  for  two  dimensions  had  been  previously 
given  by  A.  M.  Legendre.  Laplace  failed  to  make  due  acknowledg- 
ment of  this,  and  there  existed,  in  consequence,  between  the  two 
great  men,  "a  feeHng  more  than  coldness."  The  potential  function, 
V,  is  much  used  by  Laplace,  and  is  shown  by  him  to  satisfy  the  partial 

differential  equation  — o+r^+:rT=^^-     This  is  known  as  Laplace's 
^  Sa;-     Qy^     52'' 

'  A.  De  Morgan,  An  Essay  on  Probabilities,  London,  1838  (date  of  Preface) 
p.  II  of  .Appendix  I. 


264  A  HISTORY  OF  MATHEMATICS 

equation,  and  was  first  given  by  him  in  the  more  complicated  form 
which  it  assumes  in  polar  co-ordinates.  The  notion  of  potential  was, 
however,  not  introduced  into  analysis  by  Laplace.  The  honor  of 
that  achievement  belongs  to  J.  Lagrange. 

Regarding  Laplace's  equation,  P.  E.  Picard  said  in  1904:  "Few 
equations  have  been  the  object  of  so  many  works  as  this  celebrated 
equation.  The  conditions  at  the  Hmits  may  be  of  divers  forms.  The 
simplest  case  is  that  of  the  calorific  equilibrium  of  a  body  of  which 
we  maintain  the  elements  of  the  surface  at  given  temperatures;  from 
the  physical  point  of  view,  it  may  be  regarded  as  evident  that  the 
temperature,  continuous  within  the  interior  since  no  source  of  heat 
is  there,  is  determined  when  it  is  given  at  the  surface.  A  more  general 
case  is  that  where  ...  the  temperature  may  be  given  on  one  portion, 
while  there  is  radiation  on  another  portion.  These  questions  .  .  , 
have  greatly  contributed  to  the  orientation  of  the  theory  of  partial 
differential  equations.  They  have  called  attention  to  types  of  deter- 
minations of  integrals,  which  would  not  have  presented  themselves 
in  remaining  at  a  point  of  view  purely  abstract."  ^ 

Among  the  minor  discoveries  of  Laplace  are  his  method  of  solving 
equations  of  the  second,  third,  and  fourth  degrees,  his  memoir  on 
singular  solutions  of  differential  equations,  his  researches  in  finite 
differences  and  in  determinants,  the  establishment  of  the  expansion 
theorem  in  determinants  which  had  been  previously  given  by  A.  T. 
Vandermonde  for  a  special  case,  the  determination  of  the  complete 
integral  of  the  linear  differential  equation  of  the  second  order.  In 
the  Mecanique  Celeste  he  made  a  generalization  of  Lagrange's  theorem 
on  the  development  of  functions  in  series  known  as  Laplace's  theorem. 

Laplace's  investigations  in  physics  were  quite  extensive.  We  men- 
tion here  his  correction  of  Newton's  formula  on  the  velocity  of  sound 
in  gases  by  taking  into  account  the  changes  of  elasticity  due  to  the 
heat  of  compression  and  cold  of  rarefaction;  his  researches  on  the 
theory  of  tides;  his  mathematical  theory  of  capillarity;  his  explanation 
of  astronomical  refraction;  his  formulae  for  measuring  heights  by  the 
barometer. 

Laplace's  writings  stand  out  in  bold  contrast  to  those  of  J.  Lagrange 
in  their  lack  of  elegance  and  symmetry.  Laplace  looked  upon  mathe- 
matics as  the  tool  for  the  solution  of  physical  problems.  The  true 
result  being  once  reached,  he  devoted  little  time  to  explaining  the 
various  steps  of  his  analysis,  or  in  polishing  his  work.  The  last  years 
of  his  hfe  were  spent  mostly  at  Arcueil  in  peaceful  retirement  on  a 
country-place,  where  he  pursued  his  studies  with  his  usual  vigor 
until  his  death.  He  was  a  great  admirer  of  L.  Euler,  and  would  often 
say,  "Lisez  Euler,  lisez  Euler,  c'est  notre  maitre  a  tous." 

The  latter  part  of  the  eighteenth  century  brought  forth  researches 
on  the  graphic  representation  of  imaginaries,  all  of  which  remained 
'  Congress  of  Arts  and  Science,  St.  Louis,  1904,  Vol.  I,  p.  506. 


EULER,  LAGRANGE  AND  LAPLACE  265 

quite  unnoticed  at  that  time.  During  the  time  of  R.  Descartes,  I. 
Newton  and  L.  Euler,  the  negative  and  the  imaginary  came  to  be 
accepted  as  numbers,  but  the  latter  was  still  regarded  as  an  algebraic 
fiction.  A  little  over  a  hundred  years  after  J.  Wallis's  unsuccessful 
efforts  along  the  line  of  graphic  representation  of  imaginaries,  "a 
modest  scientist, "  Henri  Dominique  Truel,  pictured  imaginaries  upon 
a  line  that  was  perpendicular  to  the  line  representing  real  numbers. 
So  far  as  known,  Truel  published  nothing,  nor  are  his  manuscripts 
extant.  All  we  know  about  him  is  a  brief  reference  to  him  made  by 
A.  L.  Cauchy,^  who  says  that  Truel  had  his  graphic  scheme  as  early 
as  1786,  and  about  1810  turned  his  manuscripts  over  to  Augustin 
Normauf,  a  ship  builder  in  Havre.  W.  J.  G.  Karsten's  graphic  scheme 
of  1768  was  confined  to  imaginary  logarithms.  The  earliest  printed 
graphic  representation  of  V  —  i  and  a+b^  —  i  was  given  in  an  "  Essay 
on  the  Analytic  Representation  of  Direction,  with  Applications  in 
Particular  to  the  Determination  of  Plane  and  Spherical  Polygons" 
presented  in  1797  by  Caspar  Wessel  (i 745-1818)  to  the  Royal  Academy 
of  Sciences  and  Letters  of  Denmark  and  published  in  Vol.  V  of  its 
Memoirs  in  1799.  Wessel  was  born  in  Jonsrud,  in  Norway.  For 
many  years  he  was  in  the  employ  of  the  Danish  Academy  of  Sciences 
as  a  surveyor.  His  paper  lay  buried  in  the  Transactions  of  the  Danish 
Academy  for  nearly  a  century.  In  1897  a  French  translation  was 
brought  out  by  the  Danish  Academy.^  Another  noteworthy  publica- 
tion which  remained  unknown  for  many  years  is  an  Essay  ^  published 
in  1806  by  Jean  Robert  Argand  (i 768-1822)  of  Geneva,  containing  a 
geometric  representation  of  a+V  — 1&.  Some  parts  of  his  paper  are 
less  rigorous  than  the  corresponding  parts  of  Wessel.  Argand  gave 
some  remarkable  applications  to  trigonometry,  geometry  and  algebra. 
The  word  "modulus,"  to  represent  the  length  of  the  vector  a-\-ib, 
is  due  to  Argand.  The  writings  of  Wessel  and  Argand  being  little 
noticed,  it  remained  for  K.  F.  Gauss  to  break  down  the  last  opposition 
to  the  imaginary.  Gauss  seems  to  have  been  in  possession  of  a  graphic 
scheme  as  early  as  1799,  but  its  fuller  exposition  was  deferred  until 
1831. 

During  the  French  Revolution  the  metric  system  was  introduced. 
The  general  idea  of  decimal  subdivision  was  obtained  from  a  work  of 
Thomas  Williams,  London,  1788.  On  April  14,  1790,  Mathurin 
Jacques  Brisson  (17  23-1806)  proposed  before  the  Paris  Academy  the 
establishment  of  a  system  resting  on  a  natural  unit  of  length.  A 
scheme  was  elaborated  which  originally  included  the  decimal  sub- 
division of  the  quadrant  of  a  circle,  as  is  shown  by  the  report  made  to 

'  Cauchy,  Excrcices  d' Analyse  el  dc  phys.  math.,  T.  IV,  1847,  P-  i57- 

*  See  also  an  address  on  Wessel  by  W.  W.  Beman  in  the  Proceedings  of  the  Am. 

Ass'n  Adv.  of  Science,  Vol.  46,  1897. 

^  Imaginar\'  Quantities.     Their  Geometrical  Interpretation.    Translated  from  the 

French  of  M.  Argand  by  A.  S.  Hardy,  New  York,  i88r. 


266  A  HISTORY  OF  MATHEMATICS 

the  Academy  of  Sciences  on  March  19,  1791,  by  a  committee  con- 
sisting of  J.  C.  Borda,  J.  Lagrange,  P.  S.  Laplace,  G.  Monge,  de  Con- 
dorcet.  This  subdivision  is  found  in  the  Franqois  Callet  (i  744-1 798) 
logarithmic  tables  of  1795,  and  other  tables  published  in  France  and 
Germany.  Nevertheless  the  decimal  subdivision  of  the  quadrant  did 
not  then  prevail.^  The  commission  composed  of  Borda,  Lagrange, 
Laplace,  Monge  and  Condorcet  decided  upon  the  ten-millionth  part 
of  the  earth's  quadrant  as  the  primitive  unit  of  length.  The  length 
of  the  second's  pendulum  had  been  under  consideration,  but  was 
finally  rejected,  because  it  rested  upon  two  dissimilar  elements, 
gravity  and  time.  In  1799  the  measurement  of  the  earth's  quadrant 
was  completed  and  the  meter  estabhshed  as  the  natural  unit  of  length. 

Alexandre-Theophile  Vandermonde  (1735-1796)  studied  music 
during  his  youth  in  Paris  and  advocated  the  theory  that  all  art  rested 
upon  one  general  law,  through  which  any  one  could  become  a  com- 
poser with  the  aid  of  mathematics.  He  was  the  first  to  give  a  con- 
nected and  logical  exposition  of  the  theory  of  determinants,  and  may, 
therefore,  almost  be  regarded  as  the  founder  of  that  theory.  He  and  J. 
Lagrange  originated  the  method  of  combinations  in  solving  equations. 

Adrien  Marie  Legendre  (1752-1833)  was  educated  at  the  College 
Mazarin  in  Paris,  where  he  began  the  study  of  mathematics  under 
Abbe  Joseph  Francois  Marie  (1738-1801).  His  mathematical  genius 
secured  for  him  the  position  of  professor  of  mathematics  at  the  mili- 
tary school  of  Paris.  While  there  he  prepared  an  essay  on  the  curve 
described  by  projectiles  thrown  into  resisting  media  (ballistic  curve), 
which  captured  a  prize  offered  by  the  Royal  Academy  of  Berlin.  In 
1780  he  resigned  his  position  in  order  to  reserve  more  time  for  the 
study  of  higher  mathematics.  He  was  then  made  member  of  several 
public  commissions.  In  1795  he  was  elected  professor  at  the  Normal 
School  and  later  was  appointed  to  some  minor  government  positions. 
Owing  to  his  timidity  and  to  Laplace's  unfriendhness  toward  him,  but 
few  important  public  offices  commensurate  with  his  abiUty  were 
tendered  to  him. 

As  an  analyst,  second  only  to  P.  S.  Laplace  and  J.  Lagrange,  Legen- 
dre enriched  mathematics  by  important  contributions,  mainly  on 
elliptic  integrals,  theory  of  numbers,  attraction  of  ellipsoids,  and  least 
squares.  The  most  important  of  Legendre's  works  is  his  Fonctions 
dliptiques,  issued  in  two  volumes  in  1825  and  1826.  He  took  up  the 
subject  where  L.  Euler,  John  Landen,  and  J.  Lagrange  had  left  it, 
and  for  forty  years  was  the  only  one  to  cultivate  this  new  branch  of 
analysis,  until  at  last  C.  G.  J.  Jacobi  and  N.  H.  Abel  stepped  in  with 
admirable  new  discoveries.^    Legendre  imparted  to  the  subject  that 

^  For  details,  see  R.  Mehmke  in  Jahresb.  d.  d.  Math.  Vereinigung,  Leipzig,  1900, 
pp.  138-163. 

2  M.  Elie  de  Beaumont,  "]\Iemoir  of  Lependre."  Translated  by  C.  .\.  Alexander 
Smithsonian  Report,  1867. 


EULER,  LAGRANGE  AXD  LAPLACE  267 

connection  and  arrangement  which  belongs  to  an  independent  science. 
Starting  with  an  integral  depending  upon  the  square  root  of  a  poly- 
nomial of  the  fourth  degree  in  x,  he  showed  that  such  integrals  can  be 
brought  back  to  three  canonical  forms,  designated  by  F((f>),  E{(p),  and 
n((/)),  the  radical  being  expressed  in  the  form  A(0)  =  \/i  — ^'^  sin-0. 
He  also  undertook  the  prodigious  task  of  calculating  tables  of 
arcs  of  the  ellipse  for  different  degrees  of  amplitude  and  eccentricity, 
which  supply  the  means  of  integrating  a  large  number  of  differentials. 

An  earlier  publication  which  contained  part  of  his  researches  on 
elliptic  functions  was  his  Calcul  integral  in  three  volumes  (181 1,  1816, 
181 7),  in  which  he  treats  also  at  length  of  the  two  classes  of  definite 
integrals  named  by  him  Eiderian.  He  tabulated  the  values  of  log 
r(/>)  for  values  of  p  between  i  and  2. 

One  of  the  earliest  subjects  of  research  was  the  attraction  of  sphe- 
roids, which  suggested  to  Legendre  the  function  P„,  named  after  him. 
His  memoir  was  presented  to  the  Academy  of  Sciences  in  1783.  The 
researches  of  C.  Maclaurin  and  J.  Lagrange  suppose  the  point  at- 
tracted by  a  spheroid  to  be  at  the  surface  or  within  the  spheroid,  but 
Legendre  showed  that  in  order  to  determine  the  attraction  of  a 
spheroid  on  any  external  point  it  suffices  to  cause  the  surface  of  another 
spheroid  described  upon  the  same  foci  to  pass  through  that  point. 
Other  memoirs  on  ellipsoids  appeared  later. 

In  a  paper  of  1788  Legendre  published  criteria  for  distinguishing 
between  maxima  and  minima  in  the  calculus  of  variations,  which  were 
shown  by  J.  Lagrange  in  1797  to  be  insufficient;  this  matter  was  set 
right  by  C.  G.  J.  Jacobi  in  1836. 

The  two  household  gods  to  which  Legendre  sacrificed  with  ever- 
renewed  pleasure  in  the  silence  of  his  closet  were  the  elliptic  functions 
and  the  theory  of  numbers.  His  researches  on  the  latter  subject, 
together  with  the  numerous  scattered  fragments  on  the  theory  of 
numbers  due  to  his  predecessors  in  this  line,  were  arranged  as  far 
as  possible  into  a  systematic  whole,  and  published  in  two  large  quarto 
volumes,  entitled  Thcorie  des  nombres,  1830.  Before  the  publication 
of  this  work  Legendre  had  issued  at  divers  times  preliminary  articles. 
Its  crowning  pinnacle  is  the  theorem  of  quadratic  reciprocity,  pre- 
viously indistinctly  given  by  L.  Euler  without  proof,  but  for  the  first 
time  clearly  enunciated  and  partly  proved  by  Legendre.^ 

While  acting  as  one  of  the  commissioners  to  connect  Greenwich 
and  Paris  geodetically,  Legendre  calculated  the  geodetic  triangles  in 
France.  This  furnished  the  occasion  of  estabUshing  formulae  and 
theorems  on  geodesies,  on  the  treatment  of  the  spherical  triangle  as 
if  it  were  a  plane  triangle,  by  applying  certain  corrections  to  the 
angles,  and  on  the  method  of  least  squares,  published  for  the  first 
time  by  him  without  dernonstration  in  1806. 

^O.  Baumgart.  Ueher  das  Qiiadralische  Reciprocil'dlsgesdz,  Leipzig,  1885. 


268  A  HISTORY  OF  MATHEMATICS 

Legendre  wrote  an  Elements  de  Geomelrie,  1794,  which  enjoyed 
great  popularity,  being  generally  adopted  on  the  Continent  and  in 
the  United  States  as  a  substitute  for  Euclid.  This  great  modern  rival 
of  Euclid  passed  through  numerous  editions;  some  containing  the 
dements  of  trigonometry  and  a  proof  of  the  irrationality  of  tt  and  tt^. 
With  prophetic  vision  Legendre  remarks:  "II  est  meme  probable  que 
le  nombre  tt  n'est  pas  meme  compris  dans  les  irrationelles  algebriques, 
c'est-a-dire  qu'il  ne  peut  pas  etre  la  racine  'dune  equation  algebrique 
d'un  nombre  fini  de  termes  dont  les  coeflficients  sont  rationels." 
Much  attention  was  given  by  Legendre  to  the  subject  of  parallel  lines. 
In  the  earlier  editions  of  the  Elements,  he  made  direct  appeal  to  the 
senses  for  the  correctness  of  the  "parallel-axiom."  He  then  attempted 
to  demonstrate  that  "axiom,"  but  his  proofs  did  not  satisfy  even 
himself.  In  Vol.  XII  of  the  Memoirs  of  the  Institute  is  a  paper  by 
Legendre,  containing  his  last  attempt  at  a  solution  of  the  problem. 
Assuming  space  to  be  infinite,  he  proved  satisfactorily  that  it  is  im- 
possible for  the  sum  of  the  three  angles  of  a  triangle  to  exceed  two 
right  angles;  and  that  if  there  be  any  triangle  the  sum  of  whose  angles 
is  two  right  angles,  then  the  same  must  be  true  of  all  triangles.  But 
in  the  next  step,  to  show  that  this  sum  cannot  be  less  than  two  right 
angles,  his  demonstration  necessarily  failed.  If  it  could  be  granted 
that  the  sum  of  the  three  angles  is  always  equal  to  two  right  angles, 
then  the  theory  of  parallels  could  be  strictly  deduced. 

Another  author  who  made  contributions  to  elementary  geometry 
was  the  Italian  Lorenzo  Mascheroni  (1750-1800).  He  published  his 
Geomelria  del  compasso  (Pavia,  1797,  Palermo,  1903;  ^  French  editions 
by  A.  M.  Carette  appeared  in  1798  and  1825,  a  German  edition  by 
J.  P.  Griison  in  1825).  All  constructions  are  made  with  a  pair  of 
compasses,  but  without  restriction  to  a  fixed  radius.  He  proved  that 
all  constructions  possible  with  ruler  and  compasses  are  possible  with 
compasses  alone.  It  was  J.  V.  Poncelet  who  proved  in  1822  that  all 
such  construction  are  possible  with  ruler  alone,  if  we  are  given  a  fixed 
circle  with  its  centre  in  the  plane  of  construction;  A.  Adler  of  Vienna 
proved  in  1890  that  these  constructions  are  possible  with  ruler  alone 
whose  edges  are  parallel,  or  whose  edges  converge  in  a  point.  Masch- 
eroni claimed  that  constructions  with  compasses  are  more  accurate 
than  those  with  a  ruler.  Napoleon  proposed  to  the  French  mathe- 
maticians the  problem,  to  divide  the  circumference  of  a  circle  into 
four  equal  parts  by  the  compasses  only.  Mascheroni  does  this  by 
applying  the  radius  three  times  to  the  circumference;  he  obtains  the 
arcs  A  B,  B  C,  C  D;  then  A  D  is  a  diameter;  the  rest  is  obvious. 
E.  W.  Hobson  {Math.  Gazette,^  March  i,  1913)  and  others  have  shown 
that  all  Euclidean  constructions  can  be  carried  out  by  the  use  of 
compasses  alone. 

'  A  list  of  Mascheroni's  writings  is  given  in  L'IntermSdiaire  des  mathimaticiens, 
Vol.  iq,  1Q12,  p.  92. 


EULER,  LAGRANGE  AND  LAPLACE  269 

In  1790  Mascheroni  published  annotations  to  Euler's  Integral 
Calculus.  D'Alembert  had  argued  "le  calcul  en  defaut"  by  declaring 
that  the  astroid  x'^+yi=i  yielded  o  as  the  length  of  the  arc  from 
.r=  —  I  to  :t=  I,  y  being  taken  positive.  To  this  Mascheroni  added  in 
his  annotations  another  paradox,  by  the  contention  that,  for  .x>i, 
the  curve  is  imaginary,  yet  has  a  real  length  of  arc.^  These  paradoxes 
found  no  adequate  explanation  at  the  time,  due  to  an  inadequate 
fixing  of  the  region  of  variability. 

Joseph  Fourier  (i 768-1830)  was  born  at  Auxerre,  in  central  France. 
He  became  an  orphan  in  his  eighth  year.  Through  the  influence  of 
friends  he  was  admitted  into  the  military  school  in  his  native  place, 
then  conducted  by  the  Benedictines  of  the  Convent  of  St.  Mark.  He 
there  prosecuted  his  studies,  particularly  mathematics,  with  sur- 
prising success.  He  wished  to  enter  the  artillery,  but,  being  of  low 
birth  (the  son  of  a  tailor),  his  application  was  answered  thus:  "Fourier, 
not  being  noble,  could  not  enter  the  artillery,  although  he  were  a 
second  Newton."  -  He  was  soon  appointed  to  the  mathematical 
chair  in  the  military  school.  At  the  age  of  twenty-one  he  went  to 
Paris  to  read  before  the  Academy  of  Sciences  a  memoir  on  the  reso- 
lution of  numerical  equations,  which  was  an  improvement  on  Newton's 
method  of  approximation.  This  investigation  of  his  early  youth  he 
never  lost  sight  of.  He  lectured  upon  it  in  the  Polytechnic  School; 
he  developed  it  on  the  banks  of  the  Nile;  it  constituted  a  part  of  a 
work  entitled  Analyse  des  equationes  determines  (183 1),  which  was 
in  press  when  death  overtook  him.  This  work  contained  "Fourier's 
theorem"  on  the  number  of  real  roots  between  two  chosen  limits. 
The  French  physician  F.  D.  Budan  had  published  a  theorem  nearly 
identical  in  principle,  although  different  in  statement,  as  early  as 
1807,  but  in  1807  Budan  had  not  only  not  proved  the  theorem  known 
by  his  name,  but  had  not  yet  satisfied  himself  that  it  was  really  true. 
He  gave  a  proof  in  iSii,  which  was  printed  in  1822.  Fourier  taught 
his  theorem  to  his  pupils  in  the  Polytechnic  School  in  1796,  1797  and 
1803;  he  first  printed  the  theorem  and  its  proof  in  1820.  His  priority 
over  Budan  is  firmly  established. 

Fourier  was  anticipated  in  two  of  his  important  results.  His  im- 
provement on  the  Newton-Raphson  method  of  approximation,  render- 
ing the  process  applicable  without  the  possibility  of  failure,  was  given 
earlier  by  Mourraille,  as  was  also  Fourier's  method  of  settling  the 
question  whether  two  roots  near  the  border  line  of  equality  are  really 
equal,  or  perhaps  slightly  different,  or  perhaps  imaginary.  These 
theorems  were  eclipsed  by  that  of  Sturm,  published  in  1835. 

About  this  time  new  upper  and  lower  limits  of  the  real  roots  were 
discovered.  In  1815  Jean  Jacques  Bret  (1781-?)  professor  in  Grenoble, 
printed  three  theorems,  of  which  the  following  is  best  known:  If  frac- 

'  M.  Cantor,  op.  cit.,  Vol.  IV,  1Q08,  p.  485. 

2  D.  F.  J.  Arago,  "Joseph  Fourier,"  Smithsonian  Report,  1871. 


2/0  A  HISTORY  OF  MATHEMATICS 

tions  are  formed  by  giving  each  fraction  a  negative  coefficient  in  an 
equation  for  its  numerator,  taken  positively,  and  for  its  denominator 
the  sum  of  the  positive  coefficients  preceding  it,  if  moreover  unity  is 
added  to  each  fraction  thus  formed,  then  the  largest  number  thus 
obtainable  is  larger  than  any  root  of  the  equation.  In  1822  A.  A. 
Vene,  a  French  officer  of  engineers,  showed:  If  P  is  the  largest  negative 
coefficient,  and  if  6"  be  the  greatest  coefficient  among  the  positive 
terms  which  precede  the  first  negative  term,  then  will  P^S+i  be  a 
superior  limit. 

Fourier  took  a  prominent  part  at  his  home  in  promoting  the  Revo- 
lution. Under  the  French  Revolution  the  arts  and  sciences  seemed  for 
a  time  to  flourish.  The  reformation  of  the  weights  and  measures  was 
planned  with  grandeur  of  conception.  The  Normal  School  was 
created  in  1795,  of  which  Fourier  became  at  first  pupil,  then  lecturer. 
His  brilliant  success  secured  him  a  chair  in  the  Polytechnic  School, 
the  duties  of  which  he  afterwards  quitted,  along  with  G.  Monge  and 
C.  L.  Berthollet,  to  accompany  Napoleon  on  his  campaign  to  Egypt. 
Napoleon  founded  the  Institute  of  Egypt,  of  which  J.  Fourier  became 
secretary.  In  Egypt  he  engaged  not  only  in  scientific  work,  but  dis- 
charged important  political  functions.  After  his  return  to  France  he 
held  for  fourteen  years  the  prefecture  of  Grenoble.  During  this 
period  he  carried  on  his  elaborate  investigations  on  the  propagation  of 
heat  in  solid  bodies,  published  in  1822  in  his  work  entitled  La  Theorie 
A  nalytique  de  la  Chaleur.  This  work  marks  an  epoch  in  the  history  of 
both  pure  and  applied  mathematics.  It  is  the  source  of  all  modern 
methods  in  mathematical  physics  involving  the  integration  of  partial 
differential  equations  in  problems  where  the  boundary  values  are 
fixed  ("boundary- value  problems").  Problems  of  this  type  involve 
L.  Euler's  second  definition  of  a  "function"  in  which  the  relation  is 
not  necessarily  capable  of  being  expressed  analytically.  This  concept 
of  a  function  greatly  influenced  P.  G.  L.  Dirichlet.  The  gem  of 
Fourier's  great  book  is  "Fourier's  series."  By  this  research  a  long 
controversy  was  brought  to  a  close,  and  the  fact  recognized  that  any 
arbitrary  function  {i.  e.  any  graphically  given  function)  of  a  real 
variable  can  be  represented  by  a  trigonometric  series.  The  first 
announcement  of  this  great  discovery  was  made  by  Fourier  in  1807, 

before  the  French  Academy.    The  trigonometric  series  2/  (c»  sin  nx+ 

n  =0 

bn  cos  nx)  represents  the  function  4>{x)  for  every  value  of  x,  if  the 

coefficients  an=—  I   </)(x)  sin  nxdx,  and  bn  be  equal  to  a  similar  in- 

tegral.     The  weak  point  in  Fourier's  analysis  lies  in  his  failure  to 
jorove  generally  that  the  trigonometric  series  actually  converges  to 


EULER,  LAGRANGE  AND  LAPLACE  271 

the  value  of  the  function.  WilUam  Thomson  (later  Lord  Kelvin) 
says  that  on  May  i,  1840  (when  he  was  only  sixteen),  ''I  took  Fourier 
out  of  the  University  Library;  and  in  a  fortnight  I  had  mastered  it — 
gone  right  through  it."  Kelvin's  whole  career  was  influenced  by 
Fourier's  work  on  heat,  of  which,  he  said,  "it  is  difficult  to  say  whether 
their  uniquely  original  quality,  or  their  transcendant  interest,  or  their 
perennially  important  instructiveness  for  physical  science,  is  most 
to  be  praised."  '  Clerk  Maxwell  pronounced  it  a  great  mathematical 
poem.  Li  1827  Fourier  succeeded  P.  S.  Laplace  as  president  of  the 
council  of  the  Polytechnic  School. 

About  the  time  of  Budan  and  Fourier,  important  devices  were 
invented  in  Italy  and  England  for  the  solution  of  numerical  equations. 
The  Italian  scientific  society  in  1802  offered  a  gold  medal  for  improve- 
ments in  the  solution  of  such  equations;  it  was  awarded  in  1804  to 
Paolo  Ruifini.  With  aid  of  the  calculus  he  develops  the  theory  of 
transforming  one  equation  into  another  whose  roots  are  all  diminished 
by  a  certain  constant.'-  Then  follows  the  mechanism  for  tlie  practical 
computer,  and  here  Ruffini  has  a  device  which  is  simpler  than  Horner's 
scheme  of  1S19  and  practically  identical  with  what  is  now  known  as 
Horner's  procedure.  Horner  had  no  knowledge  of  Rufifini's  memoir. 
Nor  did  either  Horner  or  Ruffini  know  that  their  method  had  been 
given  by  the  Chinese  as  early  as  the  thirteenth  century.  Horner's 
first  paper  was  read  before  the  Royal  Society,  July  i,  1819,  and  pub- 
lished in  the  Philosophical  Transactions  for  1819.  Horner  uses  L.  F.  A. 
Arbogast's  derivatives.  The  modern  reader  is  surprised  to  find  that 
Horner's  exposition  involves  very  intricate  reasoning  which  is  in 
marked  contrast  with  the  simple  and  elementary  explanations  found 
in  modern  texts.  Perhaps  this  was  fortunate;  a  simpler  treatment 
might  have  prevented  publication  in  the  Philosophical  Transactions. 
As  it  was,  much  demur  was  made  to  the  insertion  of  the  paper.  "The 
elementary  character  of  the  subject,"  said  T.  S.  Davies,  "was  the 
professed  objection;  his  recondite  mode  of  treating  it  was  the  professed 
passport  for  its  admission."  A  second  article  of  Horner  on  his  method 
was  refused  publication  in  the  Philosophical  Transactions,  and  ap- 
peared in  1765  in  the  Mathematician,  after  the  death  of  Horner;  a 
third  article  was  printed  in  1830.  Both  Horner  and  Ruffini  explained 
their  methods  at  first  by  higher  analysis  and  later  by  elementary 
algebra;  both  offered  their  methods  as  substitutes  for  the  old  process 
of  root-extraction  of  numbers.  Ruffini's  paper  was  neglected  and 
forgotten.  Horner  was  fortunate  in  finding  two  influential  champions 
of  his  method — John  Radford  Young  (1799-1885)  of  Belfast  and  A. 
De  Morgan.  The  Rufiini-Horner  process  has  been  used  widely  in 
England  and  the  United  States,  less  widely  in  Germany,  Austria  and 

>  S.  P.  Thompson  Life  of  William  Thomson,  London,  1910,  pp.  14,  689. 
2  See  F.  Cajori,  "  Horner's  method  of  api^roximation  anticipated  by  Rulfini,"  Bull. 
Am.  Math.  Sac.  2d  S.,  Vol.  17,  1911,  pp.  409-414. 


272  A  HISTORY  OF  MATHEMATICS 

Italy,  and  not  at  all  in  France.  In  France  the  Newton-Raphson 
method  has  held  almost  undisputed  sway.^ 

Before  proceeding  to  the  origin  of  modern  geometry  we  shall  speak 
briefly  of  the  introduction  of  higher  analysis  into  Great  Britain.  This 
took  place  during  the  first  quarter  of  the  last  century.  The  British 
began  to  deplore  the  very  small  progress  that  science  was  making  in 
England  as  compared  with  its  racing  progress  on  the  Continent.  The 
first  EngUshman  to  urge  the  study  of  continental  writers  was  Robert 
Woodhouse  (1773-1827)  of  Caius  College,  Cambridge.  In  1813  the 
"Analytical  Society"  was  formed  at  Cambridge.  This  was  a  small 
club  established  by  George  Peacock,  John  Herschel,  Charles  Babbage, 
and  a  few  other  Cambridge  students,  to  promote,  as  it  was  humorously 
expressed  by  Babbage,  the  principles  of  pure  "Z)-ism,"  that  is,  the 
Leibnizian  notation  in  the  calculus  against  those  of  "dot-age,"  or 
of  the  Newtonian  notation.    This  struggle  ended  in  the  introduction 

nto  Cambridge  of  the  notation  -^,  to  the  exclusion  of  the  fluxional 

;-iotation  y.  This  was  a  great  step  in  advance,  not  on  account  of  any 
;^reat  superiority  of  the  Leibnizian  over  the  Newtonian  notation,  but 
because  the  adoption  of  the  former  opened  up  to  English  students 
the  vast  storehouses  of  continental  discoveries.  Sir  William  Thom- 
son, P.  G.  Tait,  and  some  other  modern  writers  find  it  frequently  con- 
venient to  use  both  notations.  Herschel,  Peacock,  and  Babbage 
translated,  in  1816,  from  the  French,  S.  F.  Lacroix's  briefer  treatise 
on  the  differential  and  integral  calculus,  and  added  in  1820  two 
volumes  of  examples.  Lacroix's  larger  work,  the  Traite  du  calcul 
differentiel  et  integral,  first  contained  the  term  "differential  coefficient" 
and  definitions  of  "definite"  and  "indefinite"  integrals.  It  was  one 
of  the  best  and  most  extensive  works  on  the  calculus  of  that  time. 
Of  the  three  founders  of  the  "Analytical  Society,"  Peacock  afterwards 
did  most  work  in  pure  mathematics.  Babbage  became  famous  for 
his  invention  of  a  calculating  engine  superior  to  Pascal's.  It  was 
never  finished,  owing  to  a  misunderstanding  with  the  government, 
and  a  consequent  failure  to  secure  funds.  John  Herschel,  the  eminent 
astronomer,  displayed  his  mastery  over  higher  analysis  in  memoirs 
communicated  to  the  Royal  Society  on  new  applications  of  mathe- 
matical analysis,  and  in  articles  contributed  to  cyclopaedias  on  light, 
on  meteorology,  and  on  the  history  of  mathematics.  In  the  Philo- 
sophical Transactions  of  1813  he  introduced  the  notation  sm~~\ 
tanr^x, ...  for  arcsin  x,  arctan  x,  .  .  .  He  wrote  also  log^x,  cos^x, .  . . 
for  log  (log  x),  cos  (cos  x), .  .  .,  but  in  this  notation  he  was  anticipated 
by  Heinrich  Burmann  (?-i8i7)  of  Mannheim,  a  partisan  of  the  com- 
binatory analysis  of  C.  F.  Hindenburg  in  Germany. 

1  For  references  and  further  detail,  see  Colorado  College  Publication,  General  Series 


EULER,  LAGRANGE  AND  LAPLACE  273 

George  Peacock  (i  791-1858)  was  educated  at  Trinity  College, 
Cambridge,  became  Lowndean  professor  there,  and  later,  dean  of 
Ely.  His  chief  pubHcations  are  his  Algebra,  1830  and  1842,  and  his 
Report  on  Recent  Progress  in  Analysis,  which  was  the  first  of  several 
valuable  summaries  of  scientific  progress  printed  in  the  volumes  of 
the  British  Association.  He  was  one  of  the  first  to  study  seriously 
the  fundamental  principles  of  algebra,  and  to  recognize  fully  its  purely 
symbolic  character.  He  advances,  though  somewhat  imperfectly, 
the  "principle  of  the  permanence  of  equivalent  forms."  It  assumes 
that  the  rules  applying  to  the  symbols  of  arithmetical  algebra  apply 
also  in  symbolical  algebra.  About  this  time  Duncan  Farquharson 
Gregory  (1813-1844),  fellow  of  Trinity  College,  Cambridge,  wrote 
a  paper  "on  the  real  nature  of  symbolical  algebra,"  which  brought 
out  clearly  the  commutative  and  distributive  laws.  These  laws  had 
been  noticed  years  before  by  the  inventors  of  symbolic  methods  in 
the  calculus.  It  was  F.  Servois  who  introduced  the  names  commutative 
and  distributive  in  Gergonne's  Annates,  Vol.  5,  18 14-15,  p.  93.  The 
term  associative  seems  to  be  due  to  W.  R.  Hamilton.  Peacock's 
investigations  on  the  foundation  of  algebra  were  considerably  ad- 
vanced by  A.  De  Morgan  and  H.  Hankel. 

James  Ivory  (i 765-1842)  was  a  Scotch  mathematician  who  for 
twelve  years,  beginning  in  1804,  held  the  mathematical  chair  in  the 
Royal  Military  College  at  Marlow  (now  at  Sandhurst).  He  was 
essentially  a  self-trained  mathematician,  and  almost  the  only  one  in 
Great  Britain  previous  to  the  organization  of  the  Analytical  Society 
who  was  well  versed  in  continental  mathematics.  Of  importance  is 
his  memoir  (Phil.  Trans.,  1809)  in  which  the  problem  of  the  attraction 
of  a  homogeneous  ellipsoid  upon  an  external  point  is  reduced  to  the 
simpler  problem  of  the  attraction  of  a  related  ellipsoid  upon  a  corre- 
sponding point  interior  to  it.  This  is  known  as  "Ivory's  theorem." 
He  criticised  with  undue  severity  Laplace's  solution  of  the  method 
of  least  squares,  and  gave  three  proofs  of  the  principle  without  re- 
course to  probability;  but  they  are  far  from  being  satisfactory. 

About  this  time  began  the  aggressive  investigation  of  "curves  of 
pursuit."  The  Italian  painter  Leonardo  da  Vinci  seems  to  be  the 
first  to  have  directed  attention  to  such  curves.  They  were  first 
investigated  by  Pierre  Bouguer  of  Paris  in  1732,  then  by  the  French 
collector  of  customs,  Dubois-Ayme  {Corresp.  sur  Vecole  polyt.  II,  181 1, 
p.  275)  who  stimulated  researches  carried  on  by  Thomas  de  St.  Laurent, 
Ch.  Sturm,  Jean  Joseph  Querret  and  Tedenat  {Ann.  de  Maihem.,  Vol.  13, 
1822-1823). 

By  the  researches  of  R.  Descartes  and  the  invention  of  the  calculus, 
the  analytical  treatment  of  geometry  was  brought  into  great  prom- 
inence for  over  a  century.  Notwithstanding  the  efforts  to  revive 
synthetic  methods  made  by  G.  Desargues,  B.  Pascal,  De  Lahire, 
I.  Newton,  and  C.  Maclaurin,  the  analytical  method  retained  almost 


274  A  HISTORY  OF  MATHEMATICS 

undisputed  supremacy.  It  was  reserved  for  the  genius  of  G.  Monge 
to  bring  synthetic  geometry  in  the  foreground,  and  to  open  up  new 
avenues  of  progress.  His  Geometrie  descriptive  marks  the  beginning 
of  a  wonderful  development  of  modern  geometry. 

Of  the  two  leading  problems  of  descriptive  geometry,  the  one — to 
represent  by  drawings  geometrical  magnitudes — was  brought  to  a 
high  degree  of  perfection  before  the  time  of  Monge;  the  other — to 
solve  problems  on  figures  in  space  by  constructions  in  a  plane — had 
received  considerable  attention  before  his  time.  His  most  noteworthy 
predecessor  in  descriptive  geometry  was  the  Frenchman  Amedee 
Francois  Frezier  (1682-1773).  But  it  remained  for  Monge  to  create 
descriptive  geometry  as  a  distinct  branch  of  science  by  imparting  to 
it  geometric  generality  and  elegance.  All  problems  previously  treated 
in  a  special  and  uncertain  manner  were  referred  back  to  a  few  general 
principles.  He  introduced  the  line  of  intersection  of  the  horizontal 
and  the  vertical  plane  as  the  axis  of  projection.  By  revolving  one 
plane  into  the  other  around  this  axis  or  ground-line,  many  advantages 
were  gained.^ 

Gaspard  Monge  (1746-1818)  was  born  at  Beaune.  The  construc- 
tion of  a  plan  of  his  native  town  brought  the  boy  under  the  notice  of 
a  colonel  of  engineers,  who  procured  for  him  an  appointment  in  the 
college  of  engineers  at  Mezieres.  Being  of  low  birth,  he  could  not 
receive  a  commission  in  the  army,  but  he  was  permitted  to  enter  the 
annex  of  the  school,  where  surveying  and  drawing  were  taught.  Ob- 
ser\dng  that  all  the  operations  connected  with  the  construction  of 
plans  of  fortification  were  conducted  by  long  arithmetical  processes, 
he  substituted  a  geometrical  method,  which  the  commandant  at  first 
refused  even  to  look  at;  so  short  was  the  time  in  which  it  could  be 
practised  that,  when  once  examined,  it  was  received  with  avidity. 
Monge  developed  these  methods  further  and  thus  created  his  descrip- 
tive geometry.  Owing  to  the  rivalry  between  the  French  military 
schools  of  that  time,  he  was  not  permitted  to  divulge  his  new  methods 
to  any  one  outside  of  this  institution.  In  1 768  he  was  made  professor  of 
mathematics  at  Mezieres.  In  1780,  when  conversing  with  two  of  his 
pupils,  S.  F.  LacroLx  and  S.  F.  Gay  de  Vernon  in  Paris,  he  was  obliged 
to  say,  "All  that  I  have  here  done  by  calculation,  I  could  have  done 
with  the  ruler  and  compasses,  but  I  am  not  allowed  to  reveal  these 
secrets  to  you."  But  Lacroix  set  himself  to  examine  what  the  secret 
could  be,  discovered  the  processes,  and  pubhshed  them  in  1795.  The 
method  was  published  by  Monge  himself  in  the  same  year,  first  in 
the  form  in  which  the  shorthand  writers  took  down  his  lessons  given 
at  the  Normal  School,  where  he  had  been  elected  professor,  and  then 
again,  in  revised  form,  in  the  Journal  des  ecotes  normaies.  The  next 
edition  occurred  in  1 798-1 799.  After  an  ephemeral  existence  of  only 
four  months  the  Normal  School  was  closed  in  1795.    In  the  same  year 

'  Christian  Wiener,  Lefirbuch  der  Darstelletiden  Geometric,  Leipzig,  1884,  p.  26. 


EULER,  EAC; RANGE  AxXD  LAPLACE  175 

the  Polytechnic  School  was  opened,  in  the  establishing  of  which 
Monge  took  active  part.  He  taught  there  descriptive  geometry  until 
his  departure  from  P'rance  to  accompany  Napoleon  on  the  Egyptian 
campaign.  He  was  the  first  president  of  the  Institute  of  Egxqjt. 
Monge  was  a  zealous  partisan  of  Napoleon  and  was,  for  that  reason, 
deprived  of  all  his  honors  by  Louis  XVIII.  This  and  the  destruction 
of  the  Polytechnic  School  preyed  heavily  upon  his  mind.  He  did  not 
long  survive  this  insult. 

Monge's  numerous  papers  were  by  no  means  confined  to  descriptive 
geometry.  His  analytical  discoveries  are  hardly  less  remarkable.  He 
introduced  into  analytic  geometry  the  methodic  use  of  the  equation 
of  a  line.  He  made  important  contributions  to  surfaces  of  the  second 
degree  (previously  studied  by  C.  Wren  and  L.  Euler)  and  discovered 
between  the  theory  of  surfaces  and  the  integration  of  partial  differ- 
ential equations,  a  hidden  relation  which  threw  new  light  upon  both 
subjects.  He  gave  the  differential  of  curves  of  cun.^ature,  estabhshed 
a  general  theory  of  cur\'ature,  and  applied  it  to  the  ellipsoid.  He 
found  that  the  validity  of  solutions  was  not  impaired  when  imaginaries 
are  involved  among  subsidiary  quantities.  Usually  attributed  to 
Monge  are  the  centres  of  similitude  of  circles  and  certain  theorems, 
which  were,  however,  probably  known  to  Apollonius  of  Perga.^  Monge 
published  the  following  books:  Statics,  1786;  Applications  de  Valgebre 
a  la  geometric,  1S05;  Application  de  V analyse  a  la  geonietrie.  The  last 
two  contain  most  of  his  miscellaneous  papers. 

Monge  was  an  inspiring  teacher,  and  he  gathered  around  him  a 
large  circle  of  pupils,  among  which  were  C.  Dupin,  F.  Servois,  C.  J. 
Brianchon,  Hachette,  J.  B.  Biot,  and  J.  V.  Poncelet.  Jean  Baptiste 
Biot  (1774-1862),  professor  at  the  College  de  France  in  Paris,  came  in 
contact  as  a  young  man  with  Laplace,  Lagrange,  and  Monge.  In 
1804  he  ascended  with  Gay-Lussac  in  a  balloon.  They  proved  that 
the  earth's  magnetism  is  not  appreciably  reduced  in  intensity  in 
regions  above  the  earth's  surface.  Biot  wrote  a  popular  book  on 
analytical  geometry  and  was  active  in  mathematical  physics  and 
geodesy.  He  had  a  controversy  with  Arago  who  championed  A.  J. 
Fresnel's  wave  theory  of  light.  Biot  was  a  man  of  strong  individuality 
and  great  inlluence. 

Charles  Dupin  (1784-1873),  for  many  years  professor  of  mechanics 
in  the  Conservatoire  des  Arts  et  Metiers  in  Paris,  published  in  1813 
an  important  work  on  Develop pements  de  geometi  ie,  in  which  is  intro- 
duced the  conception  of  conjugate  tangents  of  a  point  of  a  surface, 
and  of  the  indicatrix."  It  contains  also  the  theorem  known  as  "Du- 
pin's  theorem."    Surfaces  of  the  second  degree  and  descriptive  geom- 

*  R.  C.  Archibald  in  Am.  Malh.  Monthly,  Vol.  22,  1915,  pp.  6-12;  Vol.  23,  pp.  i^g- 
161. 

^  Gino  Loria,  Die  Ilmiplsdchlislcn  Theorien  der  Geoviclrie  (F.  Schiitte),  Leipzig, 
1888,  p.  49. 


276  A  HISTORY  OF  MATHEMATICS 

etry  were  successfully  studied  by  Jean  Nicolas  Pierre  Hacheite  (1769- 
1834),  who  became  professor  of  descriptive  geometry  at  the  Poly- 
technic School  after  the  departure  of  Monge  for  Rome  and  Egypt. 
In  1822  he  published  his  Traite  de  geometrie  descriptive. 

Descriptive  geometry,  which  arose,  as  we  have  seen,  in  technical 
schools  in  France,  was  transferred  to  Germany  at  the  foundation  of 
technical  schools  there.  G.  Schreiber  (1799-1871),  professor  in  Karls- 
ruhe, was  the  first  to  spread  Monge's  geometry  in  Germany  by  the 
publication  of  a  work  thereon  in  1828-1829.^  In  the  United  States 
descriptive  geometry  was  introduced  in  1816  at  the  Mihtary  Academy 
in  West  Point  by  Claude  Crozet,  once  a  pupil  at  the  Polytechnic 
School  in  Paris.    Crozet  wrote  the  first  English  work  on  the  subject.^ 

Lazare  Nicholas  Marguerite  Camot  (1753-1823)  was  born  at 
Nolay  in  Burgundy,  and  educated  in  his  native  province.  He  entered 
the  army,  but  continued  his  mathematical  studies,  and  wrote  in  1784 
a  work  on  machines,  containing  the  earUest  proof  that  kinetic  energy 
is  lost  in  collisions  of  bodies.  With  the  advent  of  the  Revolution  he 
threw  himself  into  politics,  and  when  coalesced  Europe,  in  1793, 
launched  against  France  a  million  soldiers,  the  gigantic  task  of  or- 
ganizing fourteen  armies  to  meet  the  enemy  was  achieved  by  him. 
He  was  banished  in  1796  for  opposing  Napoleon's  coup  d'etat.  The 
refugee  went  to  Geneva,  where  he  issued,  in  1797,  a  work  still  fre- 
quently quoted,  entitled.  Reflexions  sur  la  Metaphysique  du  Calcul 
Infinitesimal.  He  declared  himself  as  an  "irreconcilable  enemy  of 
kings."  After  the  Russian  campaign  he  offered  to  fight  for  France, 
though  not  for  the  empire.  On  the  restoration  he  was  exiled.  He 
died  in  Magdeburg.  His  Geometrie  de  position,  1803,  and  his  Essay  on 
Transversals,  1806,  are  important  contributions  to  modern  geometry. 
While  G.  Monge  revelled  mainly  in  three-dimensional  geometry, 
Camot  confined  himself  to  that  of  two.  By  his  effort  to  explain  the 
meaning  of  the  negative  sign  in  geometry  he  established  a  "geometry 
of  position,"  which,  however,  is  different  from  the  "Geometrie  der 
Lage"  of  to-day.  He  invented  a  class  of  general  theorems  on  pro- 
jective properties  of  figures,  which  have  since  been  pushed  to  great 
extent  by  J.  V.  Poncelet,  IMichel  Chasles,  and  others. 

Thanks  to  Carnot's  researches,  says  J.  G.  Darboux,^  "the  con- 
ceptions of  the  inventors  of  analytic  geometry,  Descartes  and  Fermat, 
retook  alongside  the  infinitesimal  calculus  of  Leibniz  and  Newton 
the  place  they  had  lost,  yet  should  never  have  ceased  to  occupy.  With 
his  geometry,  said  Lagrange,  speaking  of  Monge,  this  demon  of  a  man 
will  make  himself  immortal." 

While  in  France  the  school  of  G.  Monge  was  creating  modern 

^  C.  Wiener,  op.  cit.  p.  36. 

2  F.  Cajori,  Teaching  a>id  History  of  Mathctnalics  in  U.  S.,  Washington,  189c 
pp.  114,  117. 

*  Congress  of  Arts  and  Science,  St.  Louis,  1904,  Vol.  i,  p.  S33- 


EL'LER,  LAGRANGE  AND  LAPLACE  277 

geometry,  efforts  were  made  in  England  to  revive  Greek  geometry  by 
Robert  Simson  (1687-176S)  and  Matthew  Stewart  (1717-1785). 
Stewart  was  a  pupil  of  Simson  and  C.  Maclaurin,  and  succeeded  the 
latter  in  the  chair  at  Edinburgh.  During  the  eighteenth  century  he 
and  Maclaurin  were  the  only  prominent  mathematicians  in  Great 
Britain.  His  genius  was  ill-directed  by  the  fashion  then  prevalent  in 
England  to  ignore  higher  analysis.  In  his  Four  Tracts,  Physical  and 
Malhematical,  1761,  he  applied  geometry  to  the  solution  of  difficult 
astronomical  problems,  which  on  the  Continent  were  approached 
analytically  with  greater  success.  He  published,  in  1746,  General 
Theorems,  and  in  176,^,  his  Proposilioncs  geomctricce  more  veterum  de- 
fuonstratcB.  The  former  work  contains  sixty-nine  theorems,  of  which 
only  five  are  accompanied  by  demonstrations.  It  gives  many  inter- 
esting new  results  on  the  circle  and  the  straight  line.  Stewart  ex- 
tended some  theorems  on  transversals  due  to  Giovanni  Ceva  (1647- 
1734),  an  ItaHan,  who  pubUshed  in  1678  at  Mediolani  a  work,  De 
lineis  rectis  se  invicem  secantibus,  containing  the  theorem  now  known 
by  his  name. 


THE  NINETEENTH  AND  TWENTIETH  CENTURIES 

Introduction 

Never  more  zealously  and  successfully  has  mathematics  been 
cultivated  than  during  the  nineteenth  and  the  present  centuries.  Nor 
has  progress,  as  in  previous  periods,  been  confined  to  one  or  two 
countries.  While  the  French  and  Swiss,  who  during  the  preceding 
epoch  carried  the  torch  of  progress,  have  continued  to  develop  mathe- 
matics with  great  success,  from  other  countries  whole  armies  of  en- 
thusiastic workers  have  wheeled  into  the  front  rank.  Germany  awoke 
from  her  lethargy  by  bringing  forward  K.  F.  Gauss,  C.  G.  J.  Jacobi, 
P.  G.  L.  Dirichlet,  and  hosts  of  more  recent  men;  Great  Britain 
produced  her  A.  De  Morgan,  G.  Boole,  W.  R.  Hamilton,  A.  Cayley, 
J.  J.  Sylvester,  besides  champions  who  are  still  living;  Russia  entered 
the  arena  with  her  N.  I.  Lobachevski;  Norway  with  N.  H.  Abel; 
Italy  with  L.  Cremona;  Hungary  with  her  two  Bolyais;  the  United 
States  with  Benjamin  Peirce  and  J.  Willard  Gibbs. 

H.  S.  Wliite  of  Vassar  College  estimated  the  annual  rate  of  increase 
in  mathematical  publication  from  1870  to  1909,  and  ascertained  the 
periods  between  these  years  when  different  subjects  of  research  re- 
ceived the  greatest  emphasis.^  Taking  the  Jahrbuch  iiber  die  Fort- 
schritte  der  Mathematik,  published  since  1871  (founded  by  Carl  Ohrt- 
mann  (1839-1885)  of  the  Konigliche  Realschule  in  Berlin  and  since 
18S5  under  the  chief  editorship  of  Emil  Lampe  of  the  technische 
Hochschule  in  Berlin) ,  and  also  the  Revue  Semestrielle,  published  since 
1893  (under  the  auspices  of  the  Mathematical  Society  of  Amsterdam), 
he  counted  the  number  of  titles,  and  in  some  cases  also  the  number  of 
pages  filled  by  the  reviews  of  books  and  articles  devoted  to  a  certain 
subject  of  research,  and  reached  the  following  approximate  results: 
(i)  The  total  annual  publication  doubled  during  the  forty  years; 
(2)  During  these  forty  years,  30%  of  the  pubhcation  was  on  applied 
mathematics,  25%  on  geometry,  20%  on  analysis,  18%  on  algebra, 
7%  on  history  and  philosophy;  (3)  Geometry,  dominated  by  "Pliicker, 
his  brilliant  pupil  Klein,  Clifford,  and  Cayley,"  doubled  its  rate  of 
production  from  1870  to  1890,  then  fell  off  a  third,  to  regain  most  of 
its  loss  after  1899;  Synthetic  geometry  reached  its  maximum  in  1887 
and  then  decHned  during  the  following  twenty  years;  the  amount  of 
analytic  geometry  always  exceeded  that  of  synthetic  geometry,  the 

'  H.  S.  White,  "Forty  Years'  Fluctuations  in  Mathematical  Research,"  Science. 
K.  S.,  Vol.  42,  1915,  pp.  105-11.^. 

2  78 


INTRODUCTION 


279 


excess  being  most  pronounced  since  1887;  (4)  Analysis,  "which  takes 
its  rise  equally  from  calculus,  from  the  algebra  of  imaginaries,  from 
the  intuitions  and  the  critically  refmed  developments  of  geometry,  and 
from  abstract  logic:  the  common  servant  and  chief  ruler  of  the  other 
branches  of  mathematics,"  shows  a  trebling  in  forty  years,  reaching 
its  first  maximum  in  1890,  "probably  the  culmination  of  waves  set 
in  motion  by  Weierstrass  and  Fuchs  in  Berhn,  by  Riemann  in  Got- 
tingen,  by  Hermite  in  Paris,  Mittag-Leffler  in  Stockholm,  Dini  and 
Brioschi  in  Italy;"  before  1887  much  of  the  growth  of  analysis  is  due 
to  the  theory  of  functions  which  reaches  a  maximum  about  1887, 
with  a  sweep  of  the  curve  upward  again  after  1900,  due  to  the  theory 
of  integral  equations  and  the  influence  of  Hilbert;  (5)  Algebra,  in- 
cluding series  and  groups,  experienced  during  the  forty  years  a  steady 
gain  to  2f  times  its  original  output;  the  part  of  algebra  relating  to 
algebraic  forms,  invariants,  etc.,  reached  its  acme  before  1890  and 
then  declined  most  surprisingly;  (6)  Differential  equations  increased 
in  amount  slowly  but  steadily  from  1870,  "under  the  combined  in- 
fluence of  Weierstrass,  Darboux  and  Lie,"  showing  a  slight  decline 
in  1886,  but  "followed  by  a  marked  recovery  and  advance  during  the 
publication  of  lectures  by  Forsyth,  Picard,  Goursat  and  Painleve;" 
(7)  The  mathematical  theory  of  electricity  and  magnetism  remained 
less  than  one-fourth  of  the  whole  applied  mathematics,  but  rose  after 
1873  steadily  toward  one-fourth,  by  the  labors  of  Clerk  Maxwell, 
W.  Thomson  (Lord  Kelvin)  and  P.  G.  Tait;  (8)  The  constant  shifting 
of  mathematical  investigation  is  due  partly  to  fashion. 

The  progress  of  mathematics  has  been  greatly  accelerated  by  the 
organization  of  mathematical  societies  issuing  regular  periodicals. 
The  leading  societies  are  as  follows:  London  Mathematical  Society 
organized  in  1865,  La  sociele  mathematique  de  France  organized  in 
1872,  Edinburgh  Mathematical  Society  organized  1883,  Circolo  male- 
mat  ico  di  Palermo  organized  in  1884,  American  Mathematical  Society 
organized  in  1888  under  the  name  of  New  York  Mathematical  Society 
and  changed  to  its  present  name  in  1894,^  Deutsche  M athematiker- 
Vereinigung  organized  in  1890,  Indian  Mathematical  Society  organized 
in  1907,  Sociedad  Metematica  Espahola  organized  in  191 1,  Mathematical 
Association  of  America  organized  in  1915. 

The  number  of  mathematical  periodicals  has  enormously  increased 
during  the  passed  century.  According  to  Felix  Miiller  ^  there  were, 
up  to  1700,  only  17  periodicals  containing  mathematical  articles; 
there  were,  in  the  eighteenth  century,  210  such  periodicals,  in  the 
nineteenth  century  950  of  them. 

■  Consult  Thomas  S.  Fiske's  address  in  Bull.  Am.  Math.  Soc,  Vol.  11,  1905,  p.  238. 
Dr.  Fiske  himself  was  a  leader  in  the  organization  of  the  Society. 

-Jahrcsb.  d.  dculsch.  Malhem.  Vereinigung,  Vol.  12,  1Q03,  p.  439.  See  also  G.  A, 
Wilier  in  Historical  Introduction  to  Mathematical  Literature,  New  York,  1916, 
Chaps.  I,  II. 


28o  A  HISTORY  OF  MATHEMATICS 

A  great  stimulus  toward  mathematical  progress  have  been  the 
international  congresses  of  mathematicians.  In  1889  there  was  held 
in  Paris  a  Congres  international  de  bibliographie  des  sciences  mathe- 
matiques.  In  1893,  during  the  Columbian  Exposition,  there  was  held 
in  Chicago  an  International  Mathematical  Congress.  But,  by  com- 
mon agreement,  the  gathering  held  in  1897  ^^  Zurich,  Switzerland,  is 
called  the  "first  international  mathematical  congress."  The  second 
was  held  in  1900  at  Paris,  the  third  in  1904  at  Heidelberg,  the  fourth 
in  1908  at  Rome,  the  fifth  in  191 2  at  Cambridge  in  England.  The 
object  of  these  congresses  has  been  to  promote  friendly  relations,  to 
give  reviews  of  the  progress  and  present  state  of  different  branches  of 
mathematics,  and  to  discuss  matters  of  terminology  and  bibliography. 

One  of  the  great  co-operative  enterprises  intended  to  bring  the 
results  of  modern  research  in  digested  form  before  the  technical  reader 
is  the  Eficyklo padie  der  Mathematischen  Wissenschajten,  the  publica- 
tion of  which  was  begun  in  1898  under  the  editorship  of  Wilhelm  Franz 
Meyer  of  Konigsberg.  Prominent  as  joint  editor  was  Heinrich  Burk- 
hardt  (1861-1914)  of  Zurich,  later  of  Munich.  In  1904  was  begun  the 
publication  of  the  French  revised  and  enlarged  edition  under  the 
editorship  of  Jules  Molk  (1857-1914)  of  the  University  of  Nancy. 

As  regards  the  productiveness  of  modern  writers,  Arthur  Cayley 
said  in  1883:  ^  "It  is  difficult  to  give  an  idea  of  the  vast  extent  of 
modern  mathematics.  This  word  '  extent '  is  not  the  right  one:  I  mean 
extent  crowded  with  beautiful  detail, — not  an  extent  of  mere  uni- 
formity such  as  an  objectless  plain,  but  of  a  tract  of  beautiful  country 
seen  at  first  in  the  distance,  but  which  will  bear  to  be  rambled  through 
and  studied  in  every  detail  of  hillside  and  valley,  stream,  rock,  wood, 
and  flower."  It  is  pleasant  to  the  mathematician  to  think  that  in  his, 
as  in  no  other  science,  the  achievements  of  every  age  remain  posses- 
sions forever;  new  discoveries  seldom  disprove  older  tenets;  seldom 
is  anything  lost  or  wasted. 

If  it  be  asked  wherein  the  utility  of  some  modern  extensions  of 
mathematics  lies,  it  must  be  acknowledged  that  it  is  at  present  difficult 
to  see  how  some  of  them  are  ever  to  become  applicable  to  questions 
of  common  life  or  physical  science.  But  our  inability  to  do  this  should 
not  be  urged  as  an  argument  against  the  pursuit  of  such  studies.  In 
the  first  place,  we  know  neither  the  day  nor  the  hour  when  these 
abstract  developments  will  find  application  in  the  mechanic  arts,  in 
physical  science,  or  in  other  branches  of  mathematics.  For  example, 
the  whole  subject  of  graphical  statics,  so  useful  to  the  practical  en- 
gineer, was  made  to  rest  upon  von  Staudt's  Geometrie  der  Lage;  W.  R. 
Hamilton's  "principle  of  varying  action"  has  its  use  in  astronomy; 
complex  quantities,  general  integrals,  and  general  theorems  in  inte- 
gration offer  advantages  in  the  study  of  electricity  and  magnetism. 

1  Arthur  Cayley,  Inaugural  Address  before  the  British  Association,  1883,  Re- 
port, p.  25. 


INTRODUCTION  281 

"The  utility  of  such  researches,"  said  Spottiswoode  in  1878/  "can 
in  no  case  be  discounted,  or  even  imagined  beforehand.  Who,  for 
instance,  would  have  supposed  that  the  calculus  of  forms  or  the  theory 
of  substitutions  would  have  thrown  much  light  upon  ordinary  equa- 
tions; or  that  Abelian  functions  and  hyperelliptic  transcendents  would 
have  told  us  anything  about  the  properties  of  curves;  or  that  the 
calculus  of  operations  would  have  helped  us  in  any  way  towards  the 
figure  of  the  earth?  " 

As  a  matter  of  fact  in  the  nineteenth  century,  as  in  all  centuries, 
practical  questions  have  been  controlling  factors  in  the  growth  of 
mathematics.  Says  C.  E.  Picard:  "The  influence  of  physical  theories 
has  been  exercised  not  only  on  the  general  nature  of  the  problems  to 
be  solved,  but  even  in  the  details  of  the  analytic  transformations. 
Thus  is  currently  designated  in  recent  memoirs  on  partial  differential 
equations  under  the  name  of  Green's  formula,  a  fonnula  inspired  by 
the  primitive  formula  of  the  English  physicist.  The  theory  of  dynamic 
electricity  and  that  of  magnetism,  with  Ampere  and  Gauss,  have  been 
the  origin  of  important  progress;  the  study  of  curvilinear  integrals 
and  that  of  the  integrals  of  surfaces  have  taken  thence  all  their  de- 
velopments, and  formulas,  such  as  that  of  Stokes  which  might  also 
be  called  Ampere's  formula,  have  appeared  for  the  first  time  in  mem- 
oirs on  physics.  The  equations  on  the  propagation  of  electricity,  to 
which  are  attached  the  names  of  Ohm  and  Kirchhoff ,  while  presenting 
a  great  analogy  with  those  of  heat,  offer  often  conditions  at  the  limits 
a  little  different;  we  know  all  that  telegraphy  by  cables  owes  to  the 
profound  discussion  of  a  Fourier's  equation  carried  over  into  elec- 
tricity. The  equations  long  ago  written  of  hydrodynamics,  the 
equations  of  the  theory  of  electricity,  those  of  Maxwell  and  of  Hertz 
in  electromagnetism,  have  offered  problems  analogous  to  those  re- 
called above,  but  under  conditions  still  more  varied."  - 

Along  similar  lines  are  the  remarks  of  A.  R.  Forsyth.  In  1905  he 
said:^  "The  last  feature  of  the  century  that  will  be  mentioned  has 
been  the  increase  in  the  number  of  subjects,  apparently  dissimilar 
from  one  another,  which  are  now  being  made  to  use  mathematics  to 
some  extent.  Perhaps  the  most  surprising  is  the  application  of  mathe- 
matics to  the  domain  of  pure  thought;  this  was  effected  by  George 
Boole  in  his  treatise  'Laws  of  Thought,'  published  in  1854;  and  though 
the  developments  have  passed  considerably  beyond  Boole's  researches, 
his  work  is  one  of  those  classics  that  mark  a  new  departure.  Pohtical 
economy,  on  the  initiativ^e  of  Cournot  and  Jevons,  has  begun  to  employ 
symbols  and  to  develop  the  graphical  methods;  but  there  the  present 
use  seems  to  be  one  of  suggestive  record  and  expression,  rather  than 

'  William  Spottiswoode,  Inaugural  Address  before  British  Association,  1878, 
Report,  p.  25. 

^  Congress  of  Arts  and  Science,  St.  Louis,  1904,  Vol.  I,  pp.  507-508. 

*  Report  of  the  British  Ass'n  (South  Africa),  1905,  London,  1906,  p.  317- 


282  A  HISTORY  OF  MATKEAIATICS 

of  positive  construction.  Chemistry,  in  a  modern  spirit,  is  stretching 
out  into  mathematical  theories;  Willard  Gibbs,  in  his  memoir  on  the 
equilibrium  of  chemical  systems,  has  led  the  way;  and,  though  his 
way  is  a  path  which  chemists  find  strewn  with  the  thorns  of  analysis, 
his  work  has  rendered,  incidentally,  a  real  service  in  co-ordinating 
experimental  results  belonging  to  physics  and  to  chemistry.  A  new 
and  generahzed  theory  of  statistics  is  being  constructed;  and  a  school 
has  grown  up  which  is  applying  them  to  biological  phenomena.  Its 
activity,  however,  has  not  yet  met  with  the  sympathetic  goodwill  of 
all  the  pure  biologists;  and  those  who  remember  the  quality  of  the 
discussion  that  took  place  last  year  at  Cambridge  between  the  biome- 
tricians  and  some  of  the  biologists  will  agree  that,  if  the  new  school 
should  languish,  it  will  not  be  for  want  of  the  tonic  of  criticism." 

The  great  characteristic  of  modern  mathematics  is  its  generalizing 
tendency.  Nowadays  little  weight  is  given  to  isolated  theorems, 
says  J.  J.  Sylvester,  "except  as  affording  hints  of  an  unsuspected  new 
sphere  of  thought,  like  meteorites  detached  from  some  undiscovered 
planetary  orb  of  speculation."  In  mathematics,  as  in  all  true  sciences, 
no  subject  is  considered  in  itself  alone,  but  always  as  related  to,  or 
an  outgrowth  of,  other  things.  The  development  of  the  notion  of 
continuity  plays  a  leading  part  in  modern  research.  In  geometry 
the  principle  of  continuity,  the  idea  of  correspondence,  and  the  theory 
of  projection  constitute  the  fundamental  modern  notions.  Continuity 
asserts  itself  in  a  most  striking  way  in  relation  to  the  circular  points 
at  infinity  in  a  plane.  In  algebra  the  modern  idea  finds  expression 
in  the  theory  of  Hnear  transformations  and  invariants,  and  in  the 
recognition  of  the  value  of  homogeneity  and  symmetry. 

H.  F.  Baker  ^  said  in  1913  that,  with  the  aid  of  groups  "a  complete 
theory  of  equations  which  are  soluble  algebraically  can  be  given.  .  .  . 
But  the  theory  of  groups  has  other  applications.  .  .  .  The  group  of 
interchanges  among  four  quantities  which  leave  unaltered  the  product 
of  their  six  differences  is  exactly  similar  to  the  group  of  rotations  of  a 
regular  tetrahedron  whose  centre  is  fixed,  when  its  corners  are  inter- 
changed among  themselves.  Then  I  mention  the  historical  fact  that 
the  problem  of  ascertaining  when  that  well-known  differential  equa- 
tion called  the  hypergeometric  equation  has  all  its  solutions  expressible 
in  finite  terms  as  algebraic  functions,  was  first  solved  in  connection 
with  a  group  of  similar  kind.  For  any  linear  differential  equation  it  is 
of  primary  importance  to  consider  the  group  of  interchanges  of  its 
solutions  when  the  independent  variable,  starting  from  an  arbitrary 
point,  makes  all  possible  excursions,  returning  to  its  initial  value.  .  .  . 
There  is,  however,  a  theory  of  groups  different  from  those  so  far 
referred  to,  in  which  the  variables  can  change  continuously;  this  alone 
is  most  extensive,  as  may  be  judged  from  one  of  its  lesser  appUcations, 
the  familiar  theory  of  the  invariants  of  quantics.  Moreover,  perhaps 
'  Report  British  Ass'n  (Birmingham),  1913,  London,  19 14,  p.  371. 


INTRODUCTION  283 

the  most  masterly  of  the  analytical  discussions  of  the  theory  of 
geometry  has  been  carried  through  as  a  particular  application  of  the 
theory  of  such  groups." 

"If  the  theory  of  groups  illustrates  how  a  unifying  plan  works  in 
mathematics  beneath  the  bewildering  detail,  the  next  matter  I  refer 
to  well  shows  what  a  wealth,  what  a  grandeur,  of  thought  may  spring 
from  what  seem  slight  beginnings.  Our  ordinary  integral  calculus  is 
well-nigh  powerless  when  the  result  of  integration  is  not  expressible 
by  algebraic  or  logarithmic  functions.  The  attempt  to  extend  the 
possibilities  of  integration  to  the  case  when  the  function  to  be  inte- 
grated involves  the  square  root  of  a  polynomial  of  the  fourth  order, 
led  first,  after  many  efforts,  ...  to  the  theory  of  doubly-periodic 
functions.  To-day  this  is  much  simpler  than  ordinary  trigonometry, 
and,  even  apart  from  its  applications,  it  is  quite  incredible  that  it 
should  ever  again  pass  from  being  among  the  treasures  of  civilized 
man.  Then,  at  first  in  uncouth  form,  but  now  clothed  with  delicate 
beauty,  came  the  theory  of  general  algebraical  integrals,  of  which  the 
influence  is  spread  far  and  wide;  and  with  it  all  that  is  systematic 
in  the  theory  of  plane  curves,  and  all  that  is  associated  with  the  con- 
ception of  a  Riemann  surface.  After  this  came  the  theory  of  multiply- 
periodic  functions  of  any  number  of  variables,  which,  though  still 
very  far  indeed  from  being  complete,  has,  I  have  always  felt,  a  majesty 
of  conception  which  is  unique.  Quite  recently  the  ideas  evolved  in 
the  previous  history  have  prompted  a  vast  general  theory  of  the 
classification  of  algebraical  surfaces  according  to  their  essential  prop- 
erties, which  is  opening  endless  new  vistas  of  thought." 

The  nineteenth  century  and  the  beginning  of  the  twentieth  century 
constitute  a  period  when  the  very  foundations  of  mathematics  have 
been  re-examined  and  when  fundamental  principles  have  been  worked 
out  anew.  Says  H.  F.  Baker:  ^  "It  is  a  constantly  recurring  need  of 
science  to  reconsider  the  exact  implication  of  the  terms  employed: 
and  as  numbers  and  functions  are  inevitable  in  all  measurement,  the 
precise  meaning  of  number,  of  continuity,  of  infinity,  of  limit,  and 
so  on,  are  fundamental  questions.  .  .  .  These  notions  have  many 
pitfalls  I  may  cite.  .  .  .  the  construction  of  a  function  which  is 
continuous  at  all  points  of  a  range,  yet  possesses  no  definite  difTerential 
coefficient  at  any  point.  Are  we  sure  that  human  nature  is  the  only 
continuous  variable  in  the  concrete  world,  assuming  it  be  continuous, 
which  can  possess  such  a  vacillating  character?  .  .  .  We  could  take 
out  of  our  life  all  the  moments  at  which  we  can  say  that  our  age  is  a 
certain  number  of  years,  and  days,  and  fractions  of  day,  and  still 
have  appreciably  as  long  to  live;  this  would  be  true,  however  often, 
to  whatever  exactness,  we  named  our  age,  provided  we  were  quick 
enough  in  naming  it.  .  .  .  These  inquiries  .  .  .  have  been  associated 
also  with  the  theory  of  those  series  which  Fourier  used  so  Ijoldly,  and 
'  H.  F.  Baker,  loc.  cil.,  p.  369. 


284  A  HISTORY  OF  MATHEMATICS 

so  wickedly,  for  the  conduction  of  heat.  Like  all  discoverers,  he  took 
much  for  granted.  Precisely  how  much  is  the  problem.  This  problem 
has  led  to  the  precision  of  what  is  meant  by  a  function  of  real  variables, 
to  the  question  of  the  uniform  convergence  of  an  infinite  series,  as 
you  may  see  in  early  papers  of  Stokes,  to  new  formulation  of  the 
conditions  of  integration  and  of  the  properties  of  multiple  integrals, 
and  so  on.    And  it  remains  still  incompletely  solved. 

"Another  case  in  which  the  suggestions  of  physics  have  caused 
grave  disquiet  to  the  mathematicians  is  the  problem  of  the  variation 
of  a  definite  integral.  No  one  is  likely  to  underrate  the  grandeur  of 
the  aim  of  those  who  would  deduce  the  whole  physical  history  of  the 
world  from  the  single  principle  of  least  action.  Everyone  must  be 
interested  in  the  theorem  that  a  potential  function,  with  a  given 
value  at  the  boundary  of  a  volume,  is  such  as  to  render  a  certain  in- 
tegral, representing,  say,  the  energy,  a  minimum.  But  in  that  pro- 
portion one  desires  to  be  sure  that  the  logical  processes  employed  are 
free  from  objection.  And,  alas!  to  deal  only  with  one  of  the  earliest 
problems  of  the  subject,  though  the  finally  sufficient  conditions  for 
a  minimum  of  a  simple  integral  seemed  settled  long  ago,  and  could 
be  applied,  for  example,  to  Newton's  celebrated  problem  of  the 
solid  of  least  resistance,  it  has  since  been  shown  to  be  a  general  fact 
that  such  a  problem  cannot  have  any  definite  solution  at  all.  And, 
although  the  principle  of  Thomson  and  Dirichlet,  which  relates  to 
the  potential  problem  referred  to,  was  expounded  by  Gauss,  and 
accepted  by  Riemann,  and  remains  to-day  in  our  standard  treatise 
on  Natural  Philosophy,  there  can  be  no  doubt  that,  in  the  form  in 
which  it  was  originally  stated,  it  proves  just  nothing.  Thus  a  new 
investigation  has  been  necessary  into  the  foundations  of  the  principle. 
There  is  another  problem,  closely  connected  with  this  subject,  to 
which  I  would  allude:  the  stability  of  the  solar  system.  For  those 
who  can  make  pronouncements  in  regard  to  this  I  have  a  feeling  of 
envy;  for  their  methods,  as  yet,  I  have  a  quite  other  feeling.  The 
interest  of  this  problem  alone  is  sufficient  to  justify  the  craving  of 
the  Pure  Mathematician  for  powerful  methods  and  unexceptionable 
rigour." 

There  are  others  who  view  this  struggle  for  absolute  rigor  from  a 
different  angle.  Horace  Lamb  in  1904  spoke  as  follows:  ^  "a  traveller 
who  refuses  to  pass  over  a  bridge  until  he  has  personally  tested  the 
soundness  of  every  part  of  it  is  not  likely  to  go  very  far;  something 
must  be  risked,  even  in  Mathematics.  It  is  notorious  that  even  in 
this  realm  of  'exact'  thought,  discovery  has  often  been  in  advance  of 
strict  logic,  as  in  the  theory  of  imaginaries,  for  example,  and  in  the 
whole  province  of  analysis  of  which  Fourier's  theorem  is  a  type." 

Says  Maxime  Bocher:  ^  "There  is  what  may  perhaps  be  called  the 

'  Address  before  Section  A,  British  Ass'n,  in  Cambridge,  1904. 

*  Maxime  B6cher  in  Congress  0/  Arts  and  Science,  St.  Louis,  1904,  Vol.  I,  p.  472. 


INTRODUCTION  285 

method  of  optimism,  which  leads  us  either  willfully  or  instinctively 
to  shut  our  eyes  to  the  possibiUty  of  evil.  Thus  the  optimist  who 
treats  a  problem  in  algebra  or  analytic  geometry  will  say,  if  he  stops 
to  reflect  on  what  he  is  doing:  'I  know  that  I  have  no  right  to  divide 
by  zero;  but  there  are  so  many  other  values  which  the  expression  by 
which  I  am  dividing  might  have  that  I  will  assume  that  the  Evil 
One  has  not  thrown  a  zero  in  my  denominator  this  time.'  This 
method  .  .  .  has  been  of  great  service  in  the  rapid  development  of 
many  branches  of  mathematics." 

Definilions  of  Mathematics 

One  of  the  phases  of  the  quest  for  rigor  has  been  the  re-defining  of 
mathematics.  "Mathematics,  the  science  of  quantity"  is  an  old  idea 
which  goes  back  to  Aristotle.  A  modified  form  of  this  old  definition 
is  due  to  Auguste  Conitc  (179S-1S57),  the  French  philosopher  and 
mathematician,  the  founder  of  positivism.  Since  the  most  striking 
measurements  are  not  direct,  but  are  indirect,  as  the  determination 
of  distances  and  sizes  of  the  planets,  or  of  the  atoms,  he  defined  mathe- 
matics "the  science  of  indirect  measurement."  These  definitions 
have  been  abandoned  for  the  reason  that  several  modern  branches  of 
mathematics,  such  as  the  theory  of  groups,  analysis  situs,  projective 
geometry,  theory  of  numbers  and  the  algebra  of  logic,  have  no  relation 
to  quantity  and  measurement.  "For  one  thing,"  says  C.  J.  Keyser,^ 
"the  notion  of  the  continiimn — the  'Grand  Continuum'  as  Sylvester 
called  it — that  central  supporting  pillar  of  modern  Analysis,  has  been 
constructed  by  K.  Weierstrass,  R.  Dedekind,  Georg  Cantor  and 
others,  without  any  reference  whatever  to  quantity,  so  that  number 
and  magnitude  are  not  only  independent,  they  are  essentially  dis- 
parate." Or,  if  we  prefer  to  go  back  a  few  centuries  and  refer  to  a 
single  theorem,  we  may  quote  G.  Desargues  as  saying  that  if  the 
vertices  of  two  triangles  he  in  three  hues  meeting  in  a  point,  then  their 
sides  meet  in  three  points  lying  on  a  hne.  This  beautiful  theorem 
has  nothing  to  do  with  measurement. 

In  1S70  Benjamin  Peirce  wrote  in  his  Linear  Associative  Algebra 
that  "mathematics  is  the  science  which  draws  necessary  conclusions." 
This  definition  has  been  regarded  as  including  too  much  and  also  as 
in  need  of  elucidation  as  to  what  constitutes  a  "necessary"  conclusion. 
Reasoning  which  seemed  absolutely  conclusive  to  one  generation  no 
longer  satisfies  the  next.  According  to  present  standards  no  reasoning 
which  claims  to  be  exact  can  make  any  use  of  intuition,  but  must 
proceed  from  definitely  and  completely  stated  premises  according  to 
certain  principles  of  formal  logic. ^  Mathematical  logicians  from 
George  Boole  to  C.  S.  Peirce,  E.  Schroder,  and  G.  Peano  have  pre- 
pared the  field  so  well  that  of  late  years  Peano  and  his  followers,  and 

'  C.  J.  Keyser,  The  Human  Worth  of  Rigorous  Thinking,  New  York,  1916,  p.  277. 

*  Maxime  Bdcher,  loc.  oil.,  Vol.  I,  p   459. 


286  A  HISTORY  OF  MATHEMATICS 

independently  G.  Frege,  "have  been  able  to  make  a  rather  short 
list  of  logical  conceptions  and  principles  upon  which  it  would  seem 
that  all  exact  reasoning  depends."  But  the  validity  of  logical  prin- 
ciples must  stand  the  test  of  use,  and  on  this  point  we  may  never  be 
sure.  Frege  and  Bertrand  Russell  indej>endently  built  up  a  theory  of 
arithmetic,  each  starting  with  apparently  self-evident  logical  prin- 
ciples. Then  Russell  discovers  that  his  principles,  applied  to  a  very 
general  kind  of  logical  class,  lead  to  an  absurdity.  There  is  evident 
need  of  reconstruction  somewhere.  After  all,  are  we  merely  making 
successive  approximations  to  absolute  rigor? 

A.  B.  Kempe's  definition  is  as  follows:^  "Mathematics  is  the 
science  by  which  we  investigate  those  characteristics  of  any  subject- 
matter  of  thought  which  are  due  to  the  conception  that  it  consists 
of  a  number  of  differing  and  non-differing  individuals  and  pluralities." 
Ten  years  later  Maxime  Bocher  modified  Kempe's  definition  thus:  ^ 
"If  we  have  a  certain  class  of  objects  and  a  certain  class  of  relations, 
and  if  the  only  questions  which  we  investigate  are  whether  ordered 
groups  of  those  objects  do  or  do  not  satisfy  the  relations,  the  results 
of  the  investigation  are  called  mathematics."  Bocher  remarks  that 
if  we  restrict  ourselves  to  exact  or  deductive  mathematics,  then 
Kempe's  definition  becomes  coextensive  with  B.  Peirce's. 

Bertrand  Russell,  in  his  Principles  of  Mathematics,  Cambridge, 
1903,  regards  pure  mathematics  as  consisting  exclusively  of  deduc- 
tions "by  logical  principles  from  logical  principles."  Another  def- 
inition given  by  Russell  sounds  paradoxical,  but  really  expresses  the 
extreme  generality  and  extreme  subtleness  of  certain  parts  of  modern 
mathematics:  "Mathematics  is  the  subject  in  which  we  never  know 
what  we  are  talking  about  nor  whether  what  we  are  saying  is  true."  ^ 
Other  definitions  along  similar  lines  are  due  to  E.  Papperitz  (1892), 
G.  Itelson  (1904),  and  L.  Couturat  (1908). 

Synthetic  Geometry 

The  conflict  between  synthetic  and  analytic  methods  in  geometry 
which  arose  near  the  close  of  the  eighteenth  century  and  the  beginning 
of  the  nineteenth  has  now  come  to  an  end.  Neither  side  has  come 
out  victorious.  The  greatest  strength  is  found  to  lie,  not  in  the  sup- 
pression of  either,  but  in  the  friendly  rivalry  between  the  two,  and  in 
the  stimulating  influence  of  the  one  upon  the  other.  Lagrange  prided 
himself  that  in  his  Mecanique  Analytique  he  had  succeeded  in  avoiding 
all  figures;  but  since  his  time  mechanics  has  received  much  help  from 
geometry. 

Modern  synthetic  geometry  was  created  by  several  investigators 
about  the  same  time.    It  seemed  to  be  the  outgrowth  of  a  desire  for 

1  Proceed.  London  Math.  Soc,  Vol.  26,  1894,  p.  15. 

*  M.  B6cher,  op.  cit.,  p.  466. 

*  B.  Russell  in  International  Monthly,  Vol.  4,  1901,  p.  84. 


SYNTHETIC  GEOMETRY  287 

general  methods  which  should  serve  as  threads  of  Ariadne  to  guide 
the  student  through  the  labyrinth  of  theorems,  corollaries,  porisms, 
and  problems.  Synthetic  geometry  was  first  cultivated  by  G.  Monge, 
L.  N.  M.  Carnot,  and  J.  V.  Poncelet  in  France;  it  then  bore  rich  fruits 
at  the  hands  of  A.  F.  IVIobius  and  Jakob  Steiner  in  Germany  and 
Switzerland,  and  was  finally  developed  to  still  higher  perfection  by 
M.  Chasles  in  France,  von  Staudt  in  Germany,  and  L.  Cremona  in 
Italy. 

Jean  Victor  Poncelet  (i  788-1 S67),  a  native  of  Metz,  took  part  in 
the  Russian  campaign,  was  abandoned  as  dead  on  the  bloody  field 
of  Krasnoi,  and  taken  prisoner  to  Sara  toff.  Deprived  there  of  all 
books,  and  reduced  to  the  remembrance  of  what  he  had  learned  at 
Ihe  Lyceum  at  Metz  and  the  Polytechnic  School,  where  he  had  studied 
with  predilection  the  works  of  G.  Monge,  L.  N.  M.  Carnot,  and  C.  J. 
Brianchon,  he  began  to  study  mathematics  from  its  elements.  He 
entered  upon  original  researches  which  afterwards  made  him  illus- 
trious. While  in  prison  he  did  for  mathematics  what  Bunyan  did  for 
literature, — produced  a  much-read  work,  which  has  remained  of  great 
value  down  to  the  present  time.  He  returned  to  France  in  18 14,  and 
in  1S22  pubUshed  the  work  in  question,  entitled,  Traite  des  Proprictcs 
projectives  dcs  fi^^nres.  In  it  he  investigated  the  properties  of  figures 
w^hich  remain  unaltered  by  projection  of  the  figures.  The  projection 
is  not  effected  here  by  parallel  rays  of  prescribed  direction,  as  with 
G.  Monge,  but  by  central  projection.  Thus  perspective  projection, 
used  before  him  by  G.  Desargues,  B.  Pascal,  I.  Newton,  and  J.  H. 
Lambert,  was  elevated  by  him  into  a  fruitful  geometric  method. 
Poncelet  formulated  the  so-called  principle  of  continuity,  which  asserts 
that  properties  of  a  figure  which  hold  when  the  figure  varies  according 
to  definite  laws  wnll  hold  also  when  the  figure  assumes  some  limiting 
position. 

"Poncelet,"  says  J.  G.  Darboux,^  "could  not  content  himself  with 
the  insuflficient  resources  furnished  by  the  method  of  projections;  to 
attain  imaginaries  he  created  that  famous  principle  of  continuity 
which  gave  birth  to  such  long  discussions  between  him  and  A.  L. 
Cauchy.  Suitably  enunciated,  this  principle  is  excellent  and  can 
render  great  service.  Poncelet  was  wrong  in  refusing  to  present  it 
as  a  simple  consequence  of  analysis;  and  Cauchy,  on  the  other  hand, 
was  not  willing  to  recognize  that  his  own  objections,  apphcable  with- 
out doubt  to  certain  transcendent  figures,  were  without  force  in  the 
applications  made  by  the  author  of  the  Traite  des  proprieles  projec- 
tives.'' J.  D.  Gergonne  characterized  the  principle  as  a  valuable 
instrument  for  the  discovery  of  new  truths,  which  nevertheless  did 
not  make  stringent  proofs  superfluous.^    By  this  principle  of  geometric 

^  Congress  of  Arts  a>id  Science,  St.  Louis,  1904,  Vol.  I,  p.  539. 
*  E.  Kotter,  Die  Eniwickelung  der  synthetischen  Geometric  von  Monge  bis  auj 
Stavdt,  Leipzig,  1901,  p.  123. 


288  A  HISTORY  OF  MATHEMATICS 

continuity  Poncelet  was  led  to  the  consideration  of  points  and  lines 
which  vanish  at  infinity  or  become  imaginary.  The  inclusion  of  such 
ideal  points  and  lines  was  a  gift  which  pure  geometry  received  from 
analysis,  where  imaginary  quantities  behave  much  in  the  same  way 
as  real  ones.  Poncelet  elaborated  some  ideas  of  De  Lahire,  F.  Servois, 
and  J.  D.  Gergonne  into  a  regular  method — the  method  of  "recipro- 
cal polars."  To  him  we  owe  the  Principle  of  Duality  as  a  consequence 
of  reciprocal  polars.  As  an  independent  principle  it  is  due  to  Gergonne. 
Darboux  says  that  the  significance  of  the  principle  of  duality  which 
was  "a  httle  vague  at  first,  was  suflficiently  cleared  up  by  the  dis- 
cussions which  took  place  on  this  subject  between  J.  D.  Gergonne, 
J.  V.  Poncelet  and  J.  Pliicker."  It  had  the  advantage  of  making 
correspond  to  a  proposition  another  proposition  of  wholly  different 
aspect.  "This  was  a  fact  essentially  new.  To  put  it  in  evidence, 
Gergonne  invented  the  system,  which  since  has  had  so  much  success, 
of  memoirs  printed  in  double  columns  with  correlative  propositions 
in  juxtaposition"  (Darboux). 

Joseph  Diaz  Gergonne  (i 771-1859)  was  an  officer  of  artillery,  then 
professor  of  mathematics  at  the  lyceum  in  Nimes  and  later  professor 
at  Montpellier.  He  solved  the  Apollonian  Problem  and  claimed 
superiority  of  analytic  methods  over  the  synthetic.  Thereupon 
Poncelet  published  a  purely  geometric  solution.  Gergonne  and  Ponce- 
let carried  on  an  intense  controversy  on  the  priority  of  discovering 
the  principle  of  duality.  No  doubt,  Poncelet  entered  this  field  earlier, 
while  Gergonne  had  a  deeper  grasp  of  the  principle.  Some  geometers, 
particularly  C.  J.  Brianchon,  entertained  doubts  on  the  general  valid- 
ity of  the  principle.  The  controversy  led  to  one  new  result,  namely, 
Gergonne's  considerations  of  the  class  of  a  curve  or  surface,  as  well 
as  its  order}  Poncelet  wrote  much  on  applied  mechanics.  In  1838 
the  Faculty  of  Sciences  was  enlarged  by  his  election  to  the  chair  of 
mechanics. 

J.  G.  Darboux  says  that,  "presented  in  opposition  to  analytic 
geometry,  the  methods  of  Poncelet  were  not  favorably  received  by 
the  French  analysts.  But  such  were  their  importance  and  their 
novelty,  that  without  delay  they  aroused,  from  divers  sides,  the 
most  profound  researches."  Many  of  these  appeared  in  the  Annates 
de  mathematiques ,  published  by  J.  D.  Gergonne  at  Nimes  from  1810 
to  1831.  During  over  fifteen  years  this  was  the  only  journal  in  the 
world  devoted  exclusively  to  mathematical  researches.  Gergonne 
"collaborated,  often  against  their  will,  with  the  authors  of  the  memoirs 
sent  him,  rewrote  them,  and  sometimes  made  them  say  more  or  less 
than  they  would  have  wished.  .  .  .  Gergonne,  having  become  rector 
of  the  Academy  of  Montpellier,  was  forced  to  suspend  in  1831  the 
publication  of  his  journal.  But  the  success  it  had  obtained,  the  taste 
for  research  it  had  contributed  to  develop,  had  commenced  to  bear 
^  E.  Kotter,  op.  cit.,  pp.  160-164. 


SYNTHETIC  GEOMETRY  289 

their  fruit.  L.  A.  J.  Quetelet  had  established  in  Belgium  the  Corre- 
sporidance  mathematique  et  physique.  A.  L.  Crelle,  from  1826,  brought 
out  at  BerHn  the  tirst  sheets  of  his  celebrated  journal,  where  he  pub- 
lished the  memoirs  of  N.  H.  Abel,  of  C.  G.  J.  Jacobi,  of  J.  Steiner" 
(Darboux). 

Contemporaneous  with  J.  V.  Poncelet  was  the  German  geometer, 
Augustus  Ferdinand  Mobius  (1790-1868),  a  native  of  Schulpforta  in 
Prussia.  He  stutlied  at  Gottingen  under  K.  F.  Gauss,  also  at  Leipzig 
and  Halle.  In  Leipzig  he  became,  in  1815,  privat-docent,  the  next 
year  extraordinary  professor  of  astronomy,  and  in  1844  ordinary 
professor.  This  position  he  held  till  his  death.  The  most  important 
of  his  researches  are  on  geometry.  They  appeared  in  Crcllcs  Journal, 
and  in  his  celebrated  work  entitled  Dcr  Barycentrische  Calail,  Leipzig, 
1827,  "a  work  truly  original,  remarkable  for  the  profundity  of  its 
conceptions,  the  elegance  and  the  rigor  of  its  exposition"  (Darboux). 
As  the  name  indicates,  this  calculus  is  based  upon  properties  of  the 
centre  of  gravity.^  Thus,  that  the  point  5  is  the  centre  of  gravity  of 
weights  a,  b,  c,  d  placed  at  the  points  A,  B,  C,  D  respectively,  is  ex- 
pressed by  the  equation 

ia+b+c+d)S=aA+bB+cC+dD. 
His  calculus  is  the  beginning  of  a  quadruple  algebra,  and  contains 
the  germs  of  Grassmann's  marvellous  system.  In  designating  seg- 
ments of  lines  we  find  throughout  this  work  for  the  first  time  con- 
sistency in  the  distinction  of  positive  and  negative  by  the  order  of 
letters  AB,  BA.  Similarly  for  triangles  and  tetrahedra.  The  remark 
that  it  is  always  possible  to  give  three  points  A,  B,  C  such  weights 
a,  j8,  7  that  any  fourth  point  M  in  their  plane  will  become  a  centre  of 
mass,  led  Mobius  to  a  new  system  of  co-ordinates  in  which  the  position 
of  a  point  was  indicated  by  an  equation,  and  that  of  a  line  by  co- 
ordinates. By  this  algorithm  he  found  by  algebra  many  geometric 
theorems  expressing  mainly  invariantal  properties, — for  example,  the 
theorems  on  the  anharmonic  relation.  Mobius  wrote  also  on  statics 
and  astronomy.  He  generalized  spherical  trigonometry  by  letting 
the  sides  or  angles  of  triangles  exceed  i8o"^. 

Not  only  Mobius  but  also  H.  G.  Grassmann  discarded  the  usual 
co-ordinate  systems,  and  used  algebraic  analysis.  Later  in  the  nine- 
teenth century  and  at  the  opening  of  the  twentieth  century,  these 
ideas  were  made  use  of,  notably  by  Cyy^arissos  Stephanos  (1857-1917) 
of  the  National  University  of  Athens,  H.  Wiener,  C.  Segre,  G.  Peano, 
F.  Aschieri,  E.  Study,  C.  Burali-Forti  and  Hermann  Grassmann 
(1859-  ),  a  son  of  H.  G.  Grassmann.  Their  researches,  covering 
the  fields  of  binary  and  ternary  linear  transformations,  were  brought 
together  by  the  younger  Grassmann  into  a  treatise,  Projektive  Geome- 
trie  der  Ebene  unler  Benutzung  der  Punklrcchnung  dargesleUt,  1909. 

1  J.  W.  Gibbs,  "Multiple  Algebra,"  Proceedings  Atn.  Ass' n  for  the  Advanc.  oj 
Science,  18S6. 


2go  A  HISTORY  OF  MATHEMATICS 

Jakob  Steiner  (1796-1863),  "the  greatest  geometrician  since  the 
time  of  Euclid,"  was  born  in  IJtzendorf  in  the  Canton  of  Bern.  He 
did  not  learn  to  write  till  he  was  fourteen.  At  eighteen  he  became  a 
pupil  of  Pestalozzi.  Later  he  studied  at  Heidelberg  and  Berlin. 
When  A.  L.  Crelle  started,  in  1826,  the  celebrated  mathematical 
journal  bearing  his  name,  Steiner  and  Abel  became  leading  con- 
tributors. Through  the  influence  of  C.  G.  J.  Jacobi  and  others,  the 
chair  of  geometry  was  founded  for  him  at  Berlin  in  1834.  This  posi- 
tion he  occupied  until  his  death,  which  occurred  after  years  of  bad 
health. 

In  1832  Steiner  published  his  Systeniatische  Entwickelung  der  Ab- 
hdngigkeit  geomctrischer  Gestalten  von  einander,  "in  which  is  uncovered 
the  organism  by  which  the  most  diverse  phenomena  {Erscheinungen) 
in  the  world  of  space  are  united  to  each  other."  Here  for  the  first 
time,  is  the  principle  of  duahty  introduced  at  the  outset.  This  book 
and  von  Staudt's  lay  the  foundation  on  which  synthetic  geometry  in 
its  later  form  rested.  The  researches  of  French  mathematicians,  cul- 
minating in  the  remarkable  creations  of  G.  Monge,  J.  V.  Poncelet 
and  J.  D.  Gergonne,  suggested  a  unification  of  geometric  processes. 
This  work  of  "uncovering  the  organism  by  which  the  most  different 
forms  in  the  world  of  space  are  connected  with  each  other,"  this  ex- 
posing of  "a  small  number  of  very  simple  fundamental  relations  in 
which  the  scheme  reveals  itself,  by  which  the  whole  body  of  theorems 
can  be  logically  and  easily  developed"  was  the  task  which  Steiner 
assumed.  Says  H.  Hankel:  ^  "In  the  beautiful  theorem  that  a  conic 
section  can  be  generated  by  the  intersection  of  two  projective  pencils 
(and  the  dually  correlated  theorem  referring  to  projective  ranges), 
J.  Steiner  recognized  the  fundamental  principle  out  of  which  the 
innumerable  properties  of  these  remarkable  curves  follow,  as  it  were, 
automatically  with  playful  ease."  Not  only  did  he  fairly  complete 
the  theory  of  curves  and  surfaces  of  the  second  degree,  but  he  made 
great  advances  in  the  theory  of  those  of  higher  degrees. 

In  the  Systemaiische  Eniwickelungen  (1832)  Steiner  directed  atten- 
tion to  the  complete  figure  obtained  by  joining  in  every  possible  way 
six  points  on  a  conic  and  showed  that  in  this  hexagrammum  mysticum 
the  60  "Pascal  lines"  pass  three  by  three  through  20  points  ("Steiner 
points")  which  lie  four  by  four  upon  15  straight  lines  ("  PI  acker  lines"). 
J.  Pliicker  had  sharply  criticized  Steiner  for  an  error  that  had  crept 
into  an  earlier  statement  (1S28)  of  the  last  theorem.  Now,  Steiner 
gave  the  correct  statement,  but  without  acknowledgment  to  Pliicker. 
Further  properties  of  the  hexagrammum  mysticum  are  due  to  T.  P. 
Kirkman,  A.  Cayley  and  G.  Salmon.  The  Pascal  lines  of  three  hexa- 
gons concur  in  a  new  point  ("Kirkman  point").  There  are  60  Kirk- 
man points.  Corresponding  to  three  Pascal  lines  which  concur  in  a 
Steiner  point,  there  are  three  Kirkman  points  which  lie  upon  a  straight 
'  H.  Hankel,  Elcmciilc  dcr  Projcclivischcn  Geometric,  1875,  p.  26. 


SYNTHETIC  GEOMKFRY  291 

line  ("Cayley  line").  There  are  20  Cayley  lines  which  pass  four  by 
four  through  15  "Salmon  points."  Other  new  properties  of  the  mystic 
hexagon  were  obtained  in  1S77  by  G.  Veronese  and  L.  Cremona.^ 

In  Steiner's  hands  synthetic  geometry  made  prodigious  progress. 
New  discoveries  followed  each  other  so  rapidly  that  he  often  did  not 
take  time  to  record  tlieir  demonstrations.  In  an  article  in  Crelle's 
Journal  on  Allgemeine  Eigenschaften  Algebraischer  Ciirven  he  gives 
without  proof  theorems  which  were  declared  by  L.  O.  Hesse  to  be 
"like  Fermat's  theorems,  riddles  to  the  present  and  future  genera- 
tions." Analytical  proofs  of  some  of  them  have  been  given  since  by 
others,  but  L.  Cremona  finally  proved  them  all  by  a  synthetic  method. 
Steiner  discovered  synthetically  the  two  prominent  properties  of  a 
surface  of  the  third  order;  viz.  that  it  contains  twenty-seven  straight 
lines  and  a  pentahedron  which  has  the  double  points  for  its  vertices 
and  the  lines  of  the  Hessian  of  the  given  surface  for  its  edges.  This 
subject  will  be  discussed  more  fully  later.  Steiner  made  investigations 
by  synthetic  methods  on  maxima  and  minima,  and  arrived  at  the 
solution  of  problems  which  at  that  time  surpassed  the  analytic  power 
of  the  calculus  of  variations.  It  will  appear  later  that  his  reasoning 
on  this  topic  is  not  always  free  from  criticism. 

Steiner  generalized  Malfatti's  problem.^  Giovanni  Francesco  Mal- 
fatti  (1731-1807)  of  the  university  of  Ferrara,  in  1803,  proposed  the 
problem,  to  cut  three  cylindrical  holes  out  of  a  three-sided  prism  in 
such  a  way  that  the  cylinders  and  the  prism  have  the  same  altitude 
and  that  the  volume  of  the  cylinders  be  a  maximum.  This  problem 
was  reduced  to  another,  now  generally  known  as  Malfatti's  problem: 
to  inscribe  three  circles  in  a  triangle  so  that  each  circle  will  be  tangent  to 
two  sides  of  the  triangle  and  to  the  other  two  circles.  Malfatti  gave  an 
analytical  solution,  but  Steiner  gave  without  proof  a  construction, 
remarked  that  there  were  thirty-two  solutions,  generalized  the  problem 
})y  replacing  the  three  lines  by  three  circles,  and  solved  the  analogous 
problem  for  three  dimensions.  This  general  problem  was  solved 
analytically  by  C.  H.  Schellbach  (1809-1892)  and  A.  Cayley  and  by 
R.  F.  A  Clebsch  with  the  aid  of  the  addition  theorem  of  elliptic 
functions.^  A  simple  proof  of  Steiner's  construction  was  given  by 
A.  S.  Hart  of  Trinity  College,  Dublin,  in  1856. 

Of  interest  is  Steiner's  paper,  Ueber  die  geometrischen  Constructionen, 
ausgefiihrt  mittels  der  geraden  Linie  und  eines  festen  Kreises  (1833), 
in  which  he  shows  that  all  quadratic  constructions  can  be  effected 
with  the  aid  of  only  a  ruler,  provided  that  a  fixed  circle  is  drawn  once 
for  all.  It  was  generally  known  that  all  linear  constructions  could  be 
effected  by  the  ruler,  without  other  aids  of  any  kind.    The  case  of 

'  G.  Salmon,  Conic  Sections,  6th  Ed.,  187Q,  Notes,  p.  382. 

*  Karl  Fink,  A  Brief  History  of  Mathematics,  transl.  by  W.  W.  Beman  and  D.  E. 
Smith,  Chicago,  1900,  p.  256. 

'A.  Wittstein,  zur  Geschichte  dcs  Malfatli'schcii  Proldcms,  Nordiingen,  1878. 


292  A  HISTORY  OF  MATHEMATICS 

cubic  constructions,  calling  for  the  determination  of  three  unknown 
elements  (points)  was  worked  out  in  1868  by  Ludwig  Hermann  Kot- 
tutn  (1836-1904)  of  Bonn,  and  Stephen  Smith  of  Oxford  in  two  re- 
searches which  received  the  Steiner  prize  of  the  Berlin  Academy;  it 
was  shown  that  if  a  conic  (not  a  circle)  is  given  to  start  with,  then  all 
such  constructions  can  be  done  with  a  ruler  and  compasses.  Franz 
London  (1863-1917)  of  Breslau  demonstrated  in  1895  that  these  cubic 
constructions  can  be  effected  with  a  ruler  only,  as  soon  as  a  fixed 
cubic  curve  is  once  drawn. ^ 

F.  Biitzberger  ^  has  recently  pointed  out  that  in  an  unpublished 
manuscript,  Steiner  disclosed  a  knowledge  of  the  principle  of  inver- 
sion as  early  as  1824.  In  1847  Liouville  called  it  the  transformation 
by  reciprocal  radii.  After  Steiner  this  transformation  was  found 
independently  by  J.  Bellavitis  in  1836,  J.  W.  Stubbs  and  J.  R.  Ingram 
in  1842  and  1843,  and  by  William  Thomson  (Lord  Kelvin)  in  1845. 

Steiner's  researches  are  confined  to  synthetic  geometry.  He  hated 
analysis  as  thoroughly  as  J.  Lagrange  disliked  geometry.  Steiner's 
Gesammdte  Werke  were  published  in  Berlin  in  1881  and  1882. 

Michel  Chasles  (1793-1880)  was  born  at  Epernon,  entered  the  Poly- 
technic School  of  Paris  in  181 2,  engaged  afterwards  in  business,  which 
he  later  gave  up  that  he  might  devote  all  his  time  to  scientific  pursuits. 
In  1 84 1  he  became  professor  of  geodesy  and  mechanics  at  the  Ecole 
poly  technique;  later,  "Professeur  de  Geometrie  superieure  a  la  Faculte 
des  Sciences  de  Paris."  He  was  a  voluminous  writer  on  geometrical 
subjects.  In  1837  he  published  his  admirable  Aperqu  hislorique  sur 
Vorigine  et  le  develop panent  des  metlwdes  en  geometrie,  containing  a 
history  of  geometry  and,  as  an  appendix,  a  treatise  "sur  deux  principes 
generaux  de  la  Science."  The  Aperqu  historique  is  still  a  standard 
historical  work;  the  appendix  contains  the  general  theory  of  Homog- 
raphy  (ColUneation)  and  of  duality  (Reciprocity).  The  name  duality 
is  due  to  J.  D.  Gergonne.  Chasles  introduced  the  term  anharmonic 
ratio,  corresponding  to  the  German  Doppelverhdltniss  and  to  Clifford's 
cross-ratio.  Chasles  and  J.  Steiner  elaborated  independently  the 
modern  synthetic  or  projective  geometry.  Numerous  original  memoirs 
of  Chasles  were  published  later  in  the  Journal  de  V  Ecole  Poly  technique. 
He  gave  a  reduction  of  cubics,  different  from  Newton's  in  this,  that 
the  five  curves  from  which  all  others  can  be  projected  are  symmetrical 
with  respect  to  a  centre.  In  1864  he  began  the  publication,  in  the 
Comptcs  rendus,  of  articles  in  which  he  solves  by  his  "method  of 
characteristics"  and  the  "principle  of  correspondence"  an  immense 
number  of  problems.  He  determined,  for  instance,  the  number  of 
intersections  of  two  curv^es  in  a  plane.  The  method  of  characteristics 
contains  the  basis  of  enumerative  geometry. 

As  regards  Chasles'  use  of  imaginaries,  J.  G.  Darboux  says:  "Here, 

1  Jahresb.  d.  d.  Math.  Vereinigung,  Vol.  4,  p.  163. 

2  Bull.  Am.  Math.  Soc,  Vol.  20,  1914,  p.  414. 


SYNTHETIC  GEOMETRY  293 

his  method  was  really  new.  .  .  .  But  Chasles  introduced  imaginariea 
only  by  their  symmetric  functions,  and  consequently  would  not  have 
been  able  to  define  the  cross-ratio  of  four  elements  when  these  ceased 
CO  be  real  in  whole  or  in  part.  If  Chasles  had  been  able  to  estabhsh 
the  notion  of  the  cross-ratio  of  imaginary  elements,  a  formula  he 
gives  in  the  Gcomctrie  supcrieure  (p.  118  of  the  new  edition)  would 
have  immediately  furnished  him  that  beautiful  definition  of  angle  as 
logarithm  of  a  cross-ratio  which  enabled  E.  Laguerre,  our  regretted 
confrere,  to  give  the  complete  solution,  sought  so  long,  of  the  problem 
of  the  transformation  of  relations  which  contain  at  the  same  time  angles 
and  segments  in  homography  and  correlation."  The  application  of 
the  principle  of  correspondence  was  extended  by  A.  Cayley,  A.  Brill, 
H.  G.  Zeuthen,  H.  A.  Schwarz,  G.  H.  Halphen,  and  others.  The  full 
value  of  these  principles  of  Chasles  was  not  brought  out  until  the 
appearance,  in  1879,  of  the  Kalk'id  der  Abzahlcndcn  Geometrie  by 
Hermann  Schubert  (1848-1911)  of  Hamburg.  This  work  contains 
a  masterly  discussion  of  the  problem  of  enumerative  geometry,  viz, 
to  determine  the  number  of  points,  lines,  curves,  etc.,  of  a  system 
which  fulfil  certain  conditions.  Schubert  extended  his  enumerative 
geometry  to  ^-dimensional  space. ^  The  fundamental  principle  of 
enumerative  geometry  is  the  law  of  the  "preservation  of  the  number," 
which,  as  stated  by  Schubert,  was  found  by  E.  Study  and  by  G.  Kohn 
in  1903  to  be  not  always  valid.  The  particular  problem  examined 
by  Study  and  later  also  by  F.  Severi,  considers  the  number  of  pro- 
jectivities  of  a  line  which  transform  into  itself  a  given  group  of  four 
points.  If  the  cross-ratio  of  the  group  is  not  a  cube  root  of  —  i,  the 
number  of  projectivities  is  4,  otherwise  there  are  more.  A  recent 
book  on  this  subject  is  H.  G,  Zeuthen's  Abzdhlende  Methoden  der 
Geometrie,  1914, 

To  Chasles  we  owe  the  introduction  into  projective  geometry  of 
non-projective  properties  of  figures  by  means  of  the  infinitely  distant 
imaginary  sphero-circle.-  Remarkable  is  his  complete  solution,  in 
1846,  by  synthetic  geometry,  of  the  difficult  question  of  the  attrac- 
tion of  an  ellipsoid  on  an  external  point.  This  celebrated  problem 
was  treated  alternately  by  synthetic  and  by  analytic  methods.  Colin 
Maclaurin's  results,  obtained  synthetically,  had  created  a  sensation. 
Nevertheless,  both  A.  M.  Legendre  and  S.  D.  Poisson  expressed  the 
opinion  that  the  resources  of  the  synthetic  method  were  easily  ex- 
hausted. Poisson  solved  it  analytically  in  1835.  Then  Chasles  sur- 
prised every  one  by  his  synthetic  investigations,  based  on  the  con- 
sideration of  confocal  surfaces.  Poinsot  reported  on  the  memoir  and 
remarked  on  the  analytic  and  synthetic  methods:  "It  is  certain  that 
one  cannot  afford  to  neglect  either," 

1  Gino  Loria,  Die  Hauptsdchlichslcn  TJicorini  der  Gromdric,  iSSS,  p.  124. 
^  F.   Klein,    Vcrglcichende  Bclrachtungen  i'lber  neiiere  gcomdrische  Forsckunger. 
Eriangen   1872,  p.  12. 


294  A  HISTORY  OF  MATKEMATICS 

The  labors  of  Chasles  and  Steiner  raised  synthetic  geometry  to  an 
honored  and  respected  position  by  the  side  of  analysis. 

Karl  Georg  Christian  von  Staudt  (1798-1867)  was  born  in  Rothen- 
burg  on  the  Tauber,  and  at  his  death,  was  professor  in  Erlangen.  His 
great  works  are  the  Geometrie  der  Lage,  Niirnberg,  1847,  ^^'^  his 
Beitr'dge  zur  Geometrie  der  Lage,  1856-1860.  The  author  cut  loose 
from  algebraic  formulae  and  from  metrical  relations,  particularly  the 
anharmonic  ratio  of  J.  Steiner  and  M.  Chasles,  and  then  created  a 
geometry  of  position,  which  is  a  complete  science  in  itself,  independent 
of  all  measurements.  He  shows  that  projective  properties  of  figures 
have  no  dependence  whatever  on  measurements,  and  can  be  estab- 
lished without  any  mention  of  them.  In  his  theory  of  "throws"  or 
"Wiirfe,"  he  even  gives  a  geometrical  definition  of  a  number  in  its 
relation  to  geometry  as  determining  the  position  of  a  point.  Gustav 
Kohn  of  the  University  of  Vienna  about  1894  introduced  the  throw 
as  a  fundamental  concept  underlying  the  projective  properties  of  a 
geometric  configuration,  such  that,  according  to  a  principle  of  duality 
of  this  geometry,  throws  of  figures  appear  in  pairs  of  reciprocal  throws; 
figures  of  reciprocal  throws  form  a  complete  analogy  to  figures  of 
equal  throws.  Referring  to  Von  Staudt's  numerical  co  ordinates, 
defined  without  introducing  distance  as  a  fundamental  idea,  A.  N. 
Whitehead  said  in  1906:  "The  estabhshment  of  this  result  is  one  of 
the  triumphs  of  modern  mathematical  thought." 

The  Bcilrdge  contains  the  first  complete  and  general  theory  of 
imaginary  points,  lines,  and  planes  in  projective  geometry.  Repre- 
sentation of  an  imaginary  point  is  sought  in  the  combination  of  an 
involution  with  a  determinate  direction,  both  on  the  real  line  through 
the  point.  While  purely  projective,  von  Staudt's  method  is  inti- 
mately related  to  the  problem  of  representing  by  actual  points  and 
lines  the  imaginaries  of  analytical  geometry.  Says  Kotter:  ^  Staudt 
was  the  first  who  succeeded  "in  subjecting  the  imaginary  elements  to 
the  fundamental  theorem  of  projective  geometry,  thus  returning  to 
analytical  geometry  the  present  which,  in  the  hands  of  geometri- 
cians, had  led  to  the  most  beautiful  results."  Von  Staudt's  geometry 
of  position  was  for  a  long  time  disregarded,  mainly,  no  doubt,  because 
his  book  is  extremely  condensed.  An  impulse  to  the  study  of  this 
subject  was  given  by  Culmann,  who  rests  his  graphical  statics  upon 
the  work  of  von  Staudt.  An  interpreter  of  von  Staudt  was  at  last 
found  in  Theodor  Reye  of  Strassburg,  who  wrote  a  Geometrie  der 
Lage  in  1868. 

The  graphic  representation  of  the  imaginaries  of  analytical  geom- 
etry was  systematically  undertaken  by  C.  F.  Maximilien  Mari-e  (1819- 
1891),  who  worked,  however,  on  entirely  different  lines  from  those 
of  von  Staudt.  Another  independent  attempt  was  made  in  1893  by 
F.  H.  Loud  of  Colorado  College. 

1  E.  Kotter,  op.  cit.,  p.  123. 


SYNTHETIC  GEOMETRY  295 

Synthetic  geometry  was  studied  with  much  success  by  Luigi  Cre- 
mona (1830- 1 903),  who  was  born  in  Pavia  and  became  in  i860  pro- 
fessor of  higher  geometry  in  Bologna,  in  1866  professor  of  geometry  and 
graphical  statics  in  Milan,  in  1873  professor  of  higher  mathematics 
and  director  of  the  engineering  school  at  Rome.  He  was  influenced 
by  the  writings  of  M.  Chasles,  later  he  recognized  von  Staudt  as  the 
true  founder  of  pure  geometry.  A  memoir  of  1866  on  cubic  surfaces 
secured  half  of  the  Steiner  prize  from  Berlin,  the  other  half  being 
awarded  to  Rudolf  Sturm,  then  of  Bromberg.  Cremona  used  the 
method  of  enumeration  with  great  effect.  He  wrote  on  plane  curves, 
on  surfaces,  on  birational  transformations  of  plane  and  solid  space. 
The  birational  transformations,  the  simplest  class  of  which  is  now 
called  the  "Cremona  transformation,"  proved  of  importance,  not 
only  in  geometry,  but  in  the  analytical  theory  of  algebraic  functions 
and  integrals.  It  was  developed  more  fully  by  M.  Nother  and  others. 
H.  S.  White  comments  on  this  subject  as  follows:  ^  "Beyond  the 
linear  or  projective  transformations  of  the  plane  there  were  known 
the  quadric  inversions  of  Ludwig  Immanuel  Magnus  (1790-1S61)  of 
Berlin,  changing  lines  into  conies  through  three  fundamental  points 
and  those  exceptional  points  into  singular  Unes,  to  be  discarded. 
Cremona  described  at  once  the  highest  generalization  of  these  trans- 
formations, one-to-one  for  all  points  of  the  plane  except  a  finite  set 
of  fundamental  points.  He  found  that  it  must  be  mediated  by  a 
net  of  rational  curves;  any  two  intersecting  in  one  variable  point, 
and  in  fixed  points,  ordinary  or  multiple,  which  are  the  fundamental 
points  and  which  are  themselves  tranformed  into  singular  rational 
curves  of  the  same  orders  as  the  indices  of  multiplicity  of  the 
points.  When  the  fundamental  points  are  enumerated  by  classes 
according  to  their  several  indices,  the  set  of  class  numbers  for  the  in- 
verse transformation  is  found  to  be  the  same  as  for  the  direct,  but 
usually  related  to  different  indices.  Tables  of  such  rational  nets  of 
low  orders  were  made  out  by  L.  Cremona  and  A.  Cayley,  and  a 
wide  new  vista  seemed  opening  (such  indeed  it  was  and  is)  when  si- 
multaneously three  investigators  announced  that  the  most  general 
Cremona  transformation  is  equivalent  to  a  succession  of  quadric 
transformations  of  Magnus's  type.  This  seemed  a  climax,  and  a 
set-back  to  certain  expectations."  Cremona's  theory  of  the  trans- 
formation of  curves  and  of  the  correspondence  of  points  on  curves 
was  extended  by  him  to  three  dimensions.  There  he  showed  how  a 
great  variety  of  particular  transformations  can  be  constructed,  "but 
anything  like  a  general  theory  is  still  in  the  future."  Ruled  surfaces, 
surfaces  of  the  second  order,  space-curves  of  the  third  order,  and 
the  general  theory  of  surfaces  received  much  attention  at  his  hands. 
He  was  interested  in  map-drawing,  which  had  engaged  the  attention 
of  R.  Hooke,  G.  Mercator,  J.  Lagrange,  K.  F.  Gauss  and  others.  For 
'  Bull.  Am.  Math.  Soc,  Vol.  24,  1918,  p.  242. 


296  A  HISTORY  OF  MATHEMATICS 

a  one-one  correspondence  the  surface  must  be  unicursal,  and  this  is 
sufficient.  L.  Cremona  is  associated  with  A.  Cayley,  R.  F.  A.  Clebsch, 
M.  Nother  and  others  in  the  development  of  this  theory.^  Cremona's 
writings  were  translated  into  German  by  Maximilian  Curtze  (1S37- 
1903),  professor  at  the  gymnasium  in  Thorn,  The  Opera  matematiche 
di  Luigi  Cremona  were  brought  at  Milan  in  19 14  and  191 5. 

One  of  the  pupils  of  Cremona  was  Giovanni  Battista  Guccia  (1855- 
19 14).  He  was  born  in  Palermo  and  studied  at  Rome  under  Cremona. 
In  1889  he  became  extraordinary  professor  at  the  University  of  Pal- 
ermo, in  1894  ordinary  professor.  He  gave  much  attention  to  the 
study  of  curves  and  surfaces.  He  is  best  known  as  the  founder  in 
1884  of  the  Circolo  matematico  di  Palermo,  and  director  of  its  Rendi- 
conti.  The  society  has  become  international  and  has  been  a  powerful 
stimulus  for  mathematical  research  in  Italy, 

Karl  Culmann  (1821-1881),  professor  at  the  Polytechnicum  in  Zu- 
rich, published  an  epoch-making  work  on  Die  graphische  Statik,  Zurich, 
1864,  which  has  rendered  graphical  statics  a  great  rival  of  analytical 
statics.  Before  Culmann,  Barthelemy-Edouard  Cousinery  (1790-1851) 
a  civil  engineer  at  Paris,  had  turned  his  attention  to  the  graphical 
calculus,  but  he  made  use  of  perspective,  and  not  of  modem  geometry,- 
Culmann  is  the  first  to  undertake  to  present  the  graphical  calculus 
as  a  symmetrical  whole,  holding  the  same  relation  to  the  new  geom- 
etry that  analytical  mechanics  does  to  higher  analysis.  He  makes 
use  of  the  polar  theory  of  reciprocal  figures  as  expressing  the  relation 
between  the  force  and  the  funicular  polygons.  He  deduces  this  rela- 
tion without  leaving  the  plane  of  the  two  figures.  But  if  the  polygons 
be  regarded  as  projections  of  lines  in  space,  these  lines  may  be  treated 
as  reciprocal  elements  of  a  "Nullsystem."  This  was  done  by  Clerk 
Maxwell  in  1864,  and  elaborated  further  by  L.  Crem^ona.  The  graphi- 
cal calculus  has  been  appUed  by  0.  Mohr  of  Dresden  to  the  elastic 
line  for  continuous  spans.  Henry  T.  Eddy  (1844-  ),  then  of  the 
Rose  Polytechnic  Institute,  now  of  the  University  of  Minnesota,  gives 
graphical  solutions  of  problems  on  the  maximum  stresses  in  bridges 
under  concentrated  loads,  with  aid  of  what  he  calls  "reaction  poly- 
gons." A  standard  work,  La  Statique  graphique,  1874,  was  issued  by 
Maurice  Levy  of  Paris. 

Descriptive  geometry  [reduced  to  a  science  by  G.  Monge  in  France, 
and  elaborated  further  by  his  successors,  /.  N.  P.  Hachette,  C.  Dupin, 
Theodore  Olivier  (i 793-1853)  of  Paris,  Jules  de  la  Gournerie  of  Paris] 
was  soon  studied  also  in  other  countries.  The  French  directed  their 
attention  mainly  to  the  theory  of  surfaces  and  their  curvature;  the 
Germans  and  Swiss,  through  Guido  Schreiber  (i 799-1871)  of  Karls- 

^  Proceedings  of  the  Roy.  Sac.  of  London,  Vol.  75,  London,  1905,  pp,  277- 
279. 

2  A.  Jay  du  Bois,  Graphical  Statics,  New  York,  1875,  p.  xxxii;  M.  d'Ocagne,  Traiie 
de  Nomograi>kie.  Paris,  1899,  p.  5, 


SYNTHETIC  GEOMETRY  297 

ruhe,  Karl  Pohlke  (1810-1877)  of  Berlin/  Josef  Schlesinger  (1831- 
igoi)  of  Vienna,  and  particularly  W.  Fiedler,  interwove  projective 
and  descriptive  geometry. 

Wilhelm  Fiedler  (183  2-1 9 12),  the  son  of  a  shoe-maker  in  Chemnitz, 
Saxony,  taught  mathematics  and  mechanics  in  a  technical  school  of 
Chemnitz,  1853  to  1864,  and  studied  meanwhile  the  works  of  M. 
Chasles,  G.  Lame,  B.  de  St.-Venant,  J.  V.  Poncelet,  J.  Steiner,  J. 
Pliicker,  von  Staudt,  G.  Salmon,  A.  Cayley,  J.  J.  Sylvester.  He  was 
self-taught.  On  the  recommendacion  of  A.  F.  Mobius  he  was  awarded 
in  1859  the  degree  of  doctor  of  philosophy  by  the  University  of  Leipsic 
for  a  dissertation  on  central  projection.  At  this  time  Fiedler  made 
arrangements  with  Salmon  for  a  German  elaborated  edition  of  Sal- 
mon's Conic  Sections;  it  appeared  in  i860.  In  the  same  way  were 
brought  out  by  Friedler  Salmon's  Higher  Algebra  in  1863,  Salmon's 
Geometry  of  Three  Dimensions  in  1862,  Salmon's  Higher  Plane  Curves 
in  1873.  Iri  1S64  Fiedler  became  professor  at  the  technical  high  school 
in  Prag,  and  in  1867  at  the  Polytechnic  School  in  Zurich,  where  he 
was  active  until  his  retirement  in  1907.  The  emphasis  of  Fiedler's 
activity  was  placed  upon  descriptive  geometry.  His  Darstellende 
Geometrie,  1871,  was  brought,  in  the  third  edition,  in  organic  connec- 
tion with  V.  Staudt's  geometry  of  position.  Especially  after  the  death 
of  Culmann  in  1881,  Fiedler  was  criticised  on  pedagogic  grounds  for 
excessive  emphasis  upon  geometric  construction.  A  harmonizing 
efTort  was  the  text  on  descriptive  geometry  by  Christian  Wiener 
(1826-1896)  of  the  Polytechnic  School  in  Karlsruhe.  Of  interest  is 
Fiedler's  recognition  in  1870  of  homogeneous  co-ordinates  as  cross- 
ratios,  invariant  in  all  linear  transformations;  this  idea  had  been  ad- 
vanced in  1827  by  A.  F.  Mobius,  but  had  remained  unnoticed.'-  Fied- 
ler's Zyklographie,  1882,  contained  constructions  of  problems  on  circles 
and  spheres. 

The  intervveaving  of  projective  and  descriptive  geometry  was 
carried  on  in  Italy  by  G.  Bellavitis.  The  theory  of  shades  and  shadows 
was  first  investigated  by  the  French  writers  quoted  above,  and  in  Ger- 
many treated  most  exhaustively  by  Ludwig  Burmester  of  Munich. 

Elementary  Geometry  of  the  Triangle  and  Circle 

It  is  truly  astonishing  that  during  the  nineteenth  century  new 
theorems  should  have  been  found  relating  to  such  simple  figures  as 
the  triangle  and  circle,  which  had  been  subjected  to  such  close  exam- 
ination by  the  Greeks  and  the  long  line  of  geometers  which  followed. 
It  was  L.  Euler  who  proved  in  1765  that  the  orthocenter,  circumcenter 

1  F.  J.  Obenrauch,  Geschichte  dcr  darstdlaiden  und  projcclivcn  Geometric,  Briinn, 
1897,  pp.  350,  3S2. 

*A.  Voss,  "Wilhelm  Fiedler,"  Jahresb.  d.  d.  Math.  Vcrcinigimg,  Vol.  22,  1913 
p.  107. 


298  A  HISTORY  OF  MATHEMATICS 

and  centroid  of  a  triangle  are  collinear,  lying  on  the  "Euler  line." 
H.  C.  Gossard  of  the  University  of  Oklahoma  showed  in  igi6  that 
the  three  Euler  lines  of  the  triangles  formed  by  the  Euler  hne  and  the 
sides,  taken  by  twos,  of  a  given  triangle,  form  a  triangle  triply  per- 
spective with  the  given  triangle  and  having  the  same  Euler  line. '  Con- 
spicuous among  the  new  developments  is  the  "nine-point  circle,"  the 
discovery  of  which  has  been  erroneously  ascribed  to  Euler.  Among 
the  several  independent  discoverers  is  the  Englishman,  Benjamin 
Bevan  (?-i838)  who  proposed  in  Leybourn's  Mathematical  Repository, 
I,  18,  1804,  a  theorem  for  proof  which  practically  gives  us  the  nine- 
point  circle.  The  proof  was  supplied  to  the  Repository,  I,  Part  i, 
p.  143,  by  John  Butterworth,  who  also  proposed  a  problem,  solved  by 
himself  and  John  Whitley,  from  the  general  tenor  of  which  it  appears 
that  they  knew  the  circle  in  question  to  pass  through  all  nine  points. 
These  nine  points  are  explicitly  mentioned  by  C.  J.  Brianchon  and 
J.  V.  Poncelet  in  Gergonne's  Annates  of  1821.  In  1822,  Karl  Wilhelm 
Feuerhach  (i 800-1 S34),  professor  at  the  gymnasium  in  Erlangen, 
published  a  pamphlet  in  which  he  arrives  at  the  nine-point  circle, 
and  proves  the  theorem  known  by  his  name,  that  this  circle  touches 
the  incircle  and  the  three  excircles.  The  Germans  call  it  "  Feuerbach's 
Circle."  The  last  independent  discoverer  of  this  remarkable  circle,  so 
far  as  known,  is  T.  S.  Davies,  in  an  article  of  1S27  in  the  Philosophical 
Magazine,  II,  29-31.  Feuerbach's  theorem  was  extended  by  Andrew 
Searle  Hart  (1811-1890),  fellow  of  Trinity  College,  Dublin,  who 
showed  that  the  circles  which  touch  three  given  circles  can  be  dis- 
tributed into  sets  of  four  all  touched  by  the  same  circle. 

In  1816  August  Leopold  Crclle  published  in  Berlin  a  paper  dealing 
with  certain  properties  of  plane  triangles.  He  showed  how  to  deter- 
mine a  point  fi  inside  a  triangle,  so  that  the 
angles  (taken  in  the  same  order)  formed  by 
the  lines  joining  it  to  the  vertices  are  equal. 
In  the  adjoining  figure  the  three  marked  angles 
are  equal.  If  the  construction  is  made  so  that 
angle  0'AC  =  n'CB  =  0'BA,  then  a  second  point 
fl'  is  obtained.  The  study  of  these  new 
angles  and  new  points  led  Crelle  to  exclaim: 
"It  is  indeed  wonderful  that  so  simple  a  figure  as  the  triangle  is 
so  inexhaustible  in  properties.  How  many  as  yet  unknown  proper- 
ties of  other  figures  may  there  not  be!"  Investigations  were  made 
also  by  Karl  Friedrich  Aitdreas  Jacobi  (i 795-1855)  of  Pforta  and 
some  of  his  pupils,  but  after  his  death,  in  1855,  the  whole 
matter  was  forgotten.  In  1875  the  subject  was  again  brought  before 
the  mathematical  public  by  Henri  Brocard  (1845-  )  whose  re- 
searches were  followed  up  by  a  large  number  of  investigators  in  France, 
England  and  Germany.  Unfortunately,  the  names  of  geometricians 
which  have  been  attached  to  certain  remarkable  points,  lines  and 


SYNTHETIC  GEOx\IF;rR\' 


291 


circles  are  not  always  the  names  of  the  men  who  first  studied  their 
properties.  Thus,  we  speak  of  "Brocard  points"  and  "Brocard 
Tigles,"  but  historical  research  brought  out  the  fact,  in  1884  and  1886, 
that  these  were  the  points  and  lines  which  had  been  studied  by  A.  L. 
Crelle  and  K.  F.  A.  Jacobi.  The  "Brocard  Circle"  is  Brocard's  own 
creation.  In  the  triangle  ABC,  let  ii  and  O'  be  the  first  and  second 
•'Brocard  point."    Let  A'  be  the  intersection  of  BO  and  CO';  B'  of 


AO'  and  CO;  C'  of  BO'  and  AO.  The  circle  passing  through  A', 
B',  C  is  the  "Brocard  circle."  A'B'C  is  "Brocard's  first  triangle." 
Another  like  triangle,  A"B"C"  is  called  "Brocard's  second  triangle." 
The  points  A",  B",  C",  together  with  O,  O',  and  two  other  points, 
lie  in  the  circumference  of  the  "Brocard  circle." 

In  1S73  Emile  Lemoine  (1840-1912),  the  editor  oi  T I ntermcdiaire 
dcs  mathcmaticiens,  called  attention  to  a  particular  point  within  a  plane 
triangle  which  has  been  variously  called  the  "Lemoine  point,"  "  sym- 
median  point,"  and  "Grebe  point,"  named  after  Ernst  Wilhelm  Grebe 
(1804-1874)  of  Kassel.  If  CD  is  so  drawn 
as  to  make  angles  a  and  b  equal,  then  one 
ol  the  two  lines  AB  and  CD  is  the  anti- 
parallel  of  the  other,  with  reference  to 
the  angle  O.  Now  OE,  the  bisector  of 
AB,  is  the  median  and  OF,  the  bisector  of 
the  anti-parallel  of  AB,  is  called  the  sym- 
mcdian  (abbreviated  from  symctrique  de  la 
m'diane).  The  point  of  concurrence  of  the 
three  symmedians  in  a  triangle  is  called, 
after  Robert  Tucker  (183  2-1905)  of  University  College  School  in  Lon- 
don, the  "symmedian  point."  John  Sturgeon  Mackay  (1S43-1914)  of 
Edinburgh  has  pointed  out  that  some  of  the  properties  of  this  point, 
brought  to  light  since  1873,  were  first  discovered  previously  to  that 


300  A  HISTORY  OF  MATHEMATICS 

date.  The  anti-parallels  of  a  triangle  which  pass  through  its  sym- 
median  point,  meet  its  sides  in  six  points  which  lie  on  a  circle,  called 
the  "second  Lemoine  circle."  The  "first  Lemoine  circle"  is  a  special 
case  of  a  "Tucker  circle"  and  concentric  with  the  "Brocard  circle." 
The  "Tucker  circles"  may  be  thus  defined.  Let  DF'  =  FE'  =  ED'; 
let,  moreover,  the  following  pairs  of  fines  be  anti-parallels  to  each 
other:  AB  and  ED',  BC  and  FE',  CA  and  DF';  then  the  six  points 
D,  D',  E,  E',  F,  F',  lie  on  a  "Tucker 
circle."  Vary  the  length  of  the  equal 
anti-parallels,  and  a  family  of  "Tucker 
circles"  is  obtained.  Allied  to  these 
are  the  "Taylor  circles,"  due  to  H.  M. 
Taylor  of  Trinity  College,  Cambridge. 
Still  different  types  are  the  "Mackay 
circles,"  and  the  "Neuberg  circles" 
due  to  Joseph  Neuberg  (1840-  )  of 
Luxemburg.  A  systematic  treatise  on 
this  topic,  Die  Brocardschen  Gebilde,  was 
written  by  Albrecht  Emmerich,  Berhn,  1891.  Of  the  almost  in- 
numerable mass  of  new  theorems  on  the  triangle  and  circle,  a  great 
number  is  given  in  the  Treatise  on  the  Circle  and  the  Sphere,  Oxford, 
1916,  written  by  J.  L.  Coolidge  of  Harvard  University. 

Since  1888  E.  Lemoine  of  Paris  developed  a  system,  called  geomet- 
rographics,  for  the  purpose  of  numerically  comparing  geometric  con- 
structions with  respect  to  their  simplicity.  Coolidge  calls  these 
"the  best  known  and  least  undesirable  tests  for  the  simplicity 
of  a  geometrical  construction";  A.  Emch  declares  that  "they  are 
hardly  of  any  practical  value,  in  so  far  as  they  do  not  indicate  how  to 
simplify  a  construction  or  how  to  make  it  more  accurate." 

A  new  theorem  upon  the  circumscribed  tetraedron  was  propounded 
in  1897  by  A.  S.  Bang  and  proved  by  Joh.  Gehrke.  The  theorem  is: 
Opposite  edges  of  a  circumscribed  tetraedron  subtend  equal  angles  at 
the  points  of  contact  of  the  faces  which  contain  them.  It  has  been 
the  starting-point  for  extended  developments  by  Franz  Meyer,  J. 
Neuberg  and  H.  S.  White.^ 

Link-motion 

The  generation  of  rectilinear  motion  first  arose  as  a  practical  prob- 
lem in  the  design  of  steam  engines.  A  close  approximation  to  such 
motion  is  the  "parallel  motion"  designed  by  James  Watt  in  1784: 
In  a  freely  jointed  quadrilateral  ABCD,  with  the  side  AD  fixed,  a 
point  M  on  the  side  BC  moves  in  nearly  a  straight  line.  The  equa- 
tion of  the  curve  traced  by  M,  sometimes  called  "Watt's  curve," 
was  first  derived  by  the  French  engineer,  Francois  Marie  de  Prony 

^  Bull.  Am.  Math.  Soc,  Vol.  14,  1908,  p.  220. 


SYNTHETIC  GEOMETRY 


301 


(1755-1839);  the  curve  is  of  the  sixth  order  and  was  studied  by  Dar- 
ooux  in  1879.  A  generaUzation  of  this  curve  is  the  "three-bar  curve" 
studied  by  Samuel  Robois  in  1S76  and  Reinhold  Muller  in  1902.^ 

A  beautiful  discovery  in  hnk-motion  which  came  to  attract  a  great 
deal  of  attention  was  made  by  A.  Peaucellier,  Capitaine  du  genie  a 
Nice;  in  1864  he  proposed  in  the  Nouvelles  annales  the  problem  of 
devising  compound  compasses  for  the  generation  of  a  straight  line 
and  also  a  conic.  It  is  evident  from  his  remarks  that  he  himself  had  a 
solution.  In  1873  he  published  his  solution  in  the  same  journal. 
When  Peaucellier's  cell  came  to  be  appreciated,  he  was  awarded  the 
great  mechanical  prize  of  the  Institute  of  France,  the  "  Prbc  Montyou." 
The  generation  of  exact  rectilinear  motion  had  long  been  believed  im- 
possible. Only  recently  has  it  been  pointed  out  that  straight-line 
motion  had  been  invented  before  Peaucellier  by  another  Frenchman, 
P.  F.  Sarrus  of  Strasbourg.  He  presented  an  article  and  a  model  to 
the  Paris  Academy  of  Sciences;  the  article,  without  any  figure,  was 
published  in  1853  in  the  Comptes  Rendus  ^  of  the  academy,  and  reported 
on  by  Poncelet.  The  paper  was  entirely  forgotten  until  attention  was 
called  to  it  in  1905  by  G.  T.  Bennett  of  Emmanuel  College,  Cambridge.^ 
The  pieces  ARSB  and  ATUB  are 
each  hinged  by  three  parallel  hori- 
zontal hinges,  the  two  sets  of  hinges 
having  different  directions.  Con- 
nected thus  with  B,  A  has  a  recti- 
linear movement,  up  and  down.  In 
one  respect  Sarrus's  solution  of 
the  problem  of  rectilinear  motion  is 
more  complete  than  that  of  Peau- 
cellier; for,  while  the  Peaucellier  cell 
gives  rectiUnear  motion  only  to  a 
single  point,  Sarrus's  apparatus  gives  rectilinear  motion  to  the  whole 
piece  A.  It  was  re-invented  in  1880  by  H.  M.  Brunei  and  in  1891 
by  Archibald  Barr.  Yet  to  this  day  Sarrus's  device  appears  to  re- 
main practically  unknown. 

While  Sarrus's  device  is  three-dimensional,  that  of  Peaucellier  is 
two-dimensional.  An  independent  solution  of  straight-line  motion 
was  given  in  1871  by  the  Russian  Lipkin,  a  pupil  of  P.  Chebichev  of 
the  University  of  Petrograd.  In  1874  J.  J.  Sylvester  became  interested 
in  link-motion,  and  lectured  on  it  at  the  Royal  Institution.  Dur- 
ing the  next  few  years  several  mathematicians  worked  on  linkages. 
H.  Hart  of  Woolwich  reduced  Peaucellier's  seven  links  to  four  links. 
A  new  device  by  Sylvester  has  been  called  "Sylvester's  linkage." 

1  G.  Loria,  Ebcne  Ctirven  (F.  Schutte),  Vol.  I,  iqio,  pp.  274-279. 
^Comptes  Rendus,  Vol.  36,  pp.  1036,  11 25.     The  author's  name  is  here  spelled 
'Sarrut,"  but  R.  C.  Archibald  has  pointed  out  that  this  is  a  misprint. 
*  See  Philosoph.  Magaz.,  6.  S.,  Vol.  9,  1905,  p.  803. 


302  A  HISTORY  OF  MATHEMATICS 

The  barrister  at  law,  Alfred  Bray  Kempe  of  London  showed  in  1S76 
that  a  link-motion  can  be  found  to  describe  any  given  algebraic  curve; 
he  is  the  author  of  a  popular  booklet,  How  to  draw  a  Shaight  Line, 
London,  1877.  Other  articles  of  note  on  this  subject  were  prepared 
by  Samuel  Roberts,  Arthur  Cayley,  W.  Woolsey  Johnson,  V.  Liguine 
of  the  University  of  Odessa,  and  G.  P.  X.  Ka-nigs  of  the  Ecole  Poly- 
technique  in  Paris.  The  determination  of  the  linkage  with  minimum 
number  of  pieces  by  which  a  given  curve  can  be  described  is  still  an 
unsolved  problem. 

Parallel  Lines,  Non-Euclidean  Geometry  and  Geometry  of  n  Dimensions 

During  the  nineteenth  century  very  remarkable  generalizations  were 
made,  which  reach  to  the  very  root  of  two  of  the  oldest  branches 
of  mathematics, — elementary  algebra  and  geometry.  In  geometry 
the  axioms  have  been  searched  to  the  bottom,  and  the  con- 
clusion has  been  reached  that  the  space  defined  by  Euclid's  axioms 
is  not  the  only  possible  non-contradictory  space.  Euclid  proved 
(I,  27)  that  "if  a  straight  line  falling  on  two  other  straight  lines  make 
the  alternate  angels  equal  to  one  another,  the  two  straight  lines  shall 
be  parallel  to  one  another."  Being  unable  to  prove  that  in  every 
other  case  the  two  lines  are  not  parallel,  he  assumed  this  to  be  true  in 
what  is  now  generally  called  the  ^th  "axiom,"  by  some  the  nth  or 
the  12th  "axiom." 

Simpler  and  more  obvious  axioms  have  been  advanced  as  sub- 
stitutes. As  early  as  1663,  John  WaUis  of  Oxford  recommended:  "To 
any  triangle  another  triangle,  as  large  as  you  please,  can  be  drawn, 
which  is  similar  to  the  given  triangle."  G.  Saccheri  assumed  the  ex- 
istence of  two  similar,  unequal  triangles.  Postulates  similar  to  Wallis' 
have  been  proposed  also  by  J.  H.  Lambert,  L.  Carnot,  P.  S.  Laplace, 
J.  Delboeuf.  A.  C.  Clairaut  assumes  the  existence  of  a  rectangle; 
W.  Bolyai  postulated  that  a  circle  can  be  passed  through  any  three 
points  not  in  the  same  straight  line,  A.  M.  Legendre  that  there  existed  a 
finite  triangle  whose  angle-sum  is  two  right  angles,  J.  F.  Lorenz  and 
Legendre  that  through  every  point  within  an  angle  a  line  can  be 
drawn  interescting  both  sides,  C.  L.  Dodgson  that  in  any  circle  the 
inscribed  equilateral  quadrangle  is  greater  than  any  one  of  the  seg- 
ments which  lie  outside  it.  But  probably  the  simplest  is  the  assump- 
tion made  by  Joseph  Fenn  in  his  edition  of  Euclid's  Elements,  Dub- 
lin, 1769,  and  again  sixteen  years  later  by  William  Ludlam  (17 18-1788), 
vicar  of  Norton,  and  adopted  by  John  Playfair:  "Two  straight  lines 
which  cut  one  another  can  not  both  be  parallel  to  the  same  straight 
line."  It  is  noteworthy  ^  that  this  axiom  is  distinctly  stated  in 
Proclus's  note  to  Euclid,  I,  31. 

But  the  most  numerous  efforts  to  remove  the  supposed  defect  in 

'  T.  L.  Heath,  The  Thirteen  Books  of  Euclid's  Elements,  Vol.  I,  p.  220, 


SYNTHETIC  GEOMETRY  303 

Euclid  were  attempts  to  prove  the  parallel  postulate.  After  centuries 
of  desperate  but  fruitless  endeavor,  the  bold  idea  dawned  upon  the 
minds  of  several  mathematicians  that  a  geometry  might  be  built  up 
without  assuming  the  parallel-axiom.  While  A.  M.  Legendre  still 
endeavored  to  estabHsh  the  axiom  by  rigid  proof,  Lobachexski  brought 
out  a  publication  which  assumed  the  contradictory  of  that  axiom, 
and  which  was  the  first  of  a  series  of  articles  destined  to  clear  up  ob- 
scurities in  the  fundamental  concepts,  and  greatly  to  extend  the  field 
of  geometry. 

Nicholaus  Ivanovich  Lobachevski  (1703-1856)  was  born  at  Ala- 
karief,  in  Nizhni-Novgorod,  Russia,  studied  at  Kasan,  and  from  1827 
to  1S46  was  professor  and  rector  of  the  University  of  Kasan.  His  views 
on  the  foundation  of  geometry  were  first  set  forth  in  a  paper  laid  before 
the  physico-mathematical  department  of  the  University  of  Kasan  in 
February,  1826.  This  paper  was  never  printed  and  was  lost.  His 
earliest  publication  was  in  the  Kasan  Messenger  for  1829,  and  then  in 
the  Gelehrte  Schrijten  der  Universitat  Kasan,  1836-1838,  under  the 
title,  "New  Elements  of  Geometry,  with  a  complete  theory  of  Par- 
allels." Being  in  the  Russian  language,  the  work  remained  unknown  to 
foreigners,  but  even  at  home  it  attracted  no  notice.  In  1840  he  pub- 
lished a  brief  statement  of  his  researches  in  Berlin,  under  the  title, 
Geometrische  Untersuchungen  ziir  Theorie  der  ParaUelUnien.  Loba- 
chevski constructed  an  "imaginary  geometry,"  as  he  called  it,  which 
has  been  described  by  W.  K.  Clifford  as  "quite  simple,  merely  Euclid 
without  the  vicious  assumption."  A  remarkable  part  of  this  geometry 
is  this,  that  through  a  point  an  indefinite  number  of  lines  can  be  drawn 
in  a  plane,  none  of  which  cut  a  given  line  in  the  same  plane.  A  similar 
system  of  geometry  was  deduced  independently  by  the  Bolyais  in 
Hungary,  who  called  it  "absolute  geometry." 

Wolfgang  Bolyai  de  Bolya  (i 775-1856)  was  born  in  Szekler-Land, 
Transylvania.  After  studying  at  Jena,  he  went  to  Gottingen,  where 
he  became  intimate  with  K.  F.  Gauss,  then  nineteen  years  old.  Gauss 
used  to  say  that  Bolyai  was  the  only  man  who  fully  understood  his 
views  on  the  metaphysics  of  mathematics.  Bolyai  became  professor 
at  the  Reformed  College  of  ]\Iaros-Vasarhely,  where  for  forty-seven 
years  he  had  for  his  pupils  most  of  the  later  professors  of  Transyl- 
vania. The  first  publications  of  this  remarkable  genius  were  dramas 
and  poetry.  Clad  in  old-time  planter's  garb,  he  was  truly  original  in 
his  private  life  as  well  as  in  his  mode  of  thinking.  He  was  extremely 
modest.  No  monument,  said  he,  should  stand  over  his  grave,  only  an 
apple-tree,  in  memory  of  the  three  apples;  the  two  of  Eve  and  Paris, 
which  made  hell  out  of  earth,  and  that  of  I.  Newton,  which  elevated 
the  earth  again  into  the  circle  of  heavenly  bodies.^    His  son,  Johann 

1  F.  Schmidt,  "Aus  dem  Leben  zweier  ungarischer  Mathematiker  Johann  und 
Wolfgang  Bolyai  von  Bolya,"  Grunerfs  Archiv,  48:2,  1S68.  Fran^  Schmidt  (1827- 
1901)  was  an  architect  in  Budapest. 


304  A  HISTORY  OF  MATHEMATICS 

Bolyai  (1802-1860),  was  educated  for  the  army,  and  distinguished 
himself  as  a  profound  mathematician,  an  impassioned  violin-playei. 
and  an  expert  fencer.  He  once  accepted  the  challenge  of  thirteen 
officers  on  condition  that  after  each  duel  he  might  play  a  piece  on  his 
violin,  and  he  vanquished  them  all. 

The  chief  mathematical  work  of  Wolfgang  Bolyai  appeared  in 
two  volumes,  1832-1833,  entitled  Tentamen  juventiitem  studiosam  in 
elementa  matheseos  puree  .  .  .  introducendi.  It  is  followed  by  an  ap- 
pendix composed  by  his  son  Johann.  Its  twenty-sLx  pages  make  the 
name  of  Johann  Bolyai  immortal.  He  published  nothing  else,  but  he 
left  behind  one  thousand  pages  of  manuscript. 

WTiile  Lobachevski  enjoys  priority  of  publication,  it  may  be  that 
Bolyai  developed  his  system  somewhat  earlier.  Bolyai  satisfied  him- 
self of  the  non-contradictory  character  of  his  new  geometry  on  or  be- 
fore 1825;  there  is  some  doubt  whether  Lobachevski  had  reached  this 
point  in  1826.  Johann  Bolyai's  father  seems  to  have  been  the  only 
person  in  Hungary  who  really  appreciated  the  merits  of  his  son's 
work.  For  thirty-five  years  this  appendix,  as  also  Lobachevski's 
researches,  remained  in  almost  entire  oblivion.  Finally  Richard 
Baltzer  of  the  University  of  Giessen,  in  1867,  called  attention  to  the 
wonderful  researches. 

In  1866  J.  Hoiiel  translated  Lobachevski's  Geometrische  Unter- 
suchungen  into  French.  In  1867  appeared  a  French  translation  of 
Johann  Bolyai's  Appendix.  In  1891  George  Bruce  Halsted,  then  of 
the  University  of  Texas,  rendered  these  treatises  easily  accessible  to 
American  readers  by  translations  brought  out  under  the  titles  of 
J.  Bolyai's  The  Science  Absolute  of  Space  and  N.  Lobachevski's  Geo- 
metrical Researches  on  the  Theory  of  Parallels  of  1840. 

The  Russian  and  Hungarian  mathematicians  were  not  the  only 
ones  to  whom  pangeometry  suggested  itself.  A  copy  of  the  Tentamen 
reached  K.  F.  Gauss,  the  elder  Bolyai's  former  roommate  at  Gottingen, 
and  this  Nestor  of  German  mathematicians  was  surprised  to  discover 
in  it  worked  out  what  he  himself  had  begun  long  before,  only  to  leave 
it  after  him  in  his  papers.  As  early  as  1 792  he  had  started  on  researches 
of  that  character.  His  letters  show  that  in  1799  he  was  trying  to  prove 
a  priori  the  reality  of  Euclid's  system;  but  some  time  within  the  next 
thirty  years  he  arrived  at  the  conclusion  reached  by  Lobachevski 
and  Bolyai.  In  1829  he  wrote  to  F.  W.  Bessel,  stating  that  his  "con- 
viction that  we  cannot  found  geometry  completely  a  priori  has  be- 
come, if  possible,  still  firmer,"  and  that  "if  number  is  merely  a  product 
of  our  mind,  space  has  also  a  reality  beyond  our  mind  of  which  we 
cannot  fully  foreordain  the  laws  a  priori J^  The  term  no n- Euclidean 
geometry  is  due  to  Gauss.  It  is  surprising  that  the  first  glimpses  of 
non-Euclidean  geometry  were  had  in  the  eighteenth  centur>'.  Geron- 
nimo  Saccheri  (1667-1733),  a  Jesuit  father  of  Milan,  in  1733  wrote 


SYNTHETIC  GEOMETRY  305 

Euclides  ah  omni  naevo  vindicatus  ^  (Euclid  vindicated  from  every 
flaw).  Starting  with  two  equal  lines  AC  and  BD,  drawn  perpendicu- 
lar to  a  line  AB  and  on  the  same  side  of  it,  and  joining  C  and  D,  he 
proves  that  the  angles  at  C  and  D  are  equal.  These  angles  must  be 
either  right,  or  obtuse  or  acute.  The  hypothesis  of  an  obtuse  angle 
is  demolished  by  showing  that  it  leads  to  results  in  conflict  with 
Euclid  1,17:  Any  two  angles  of  a  triangle  are  together  less  than  two 
right  angles.  The  hypothesis  of  the  acute  angle  leads  to  a  long  pro- 
cession of  theorems,  of  which  the  one  declaring  that  two  lines  which 
meet  in  a  point  at  infinity  can  be  perpendicular  at  that  point  to  the 
same  straight  line,  is  considered  contrary  to  the  nature  of  the  straight 
line;  hence  the  hypothesis  of  the  acute  angle  is  destroyed.  Though 
not  altogether  satisfied  with  his  proof,  he  declared  Euclid  "vindi- 
cated." Another  early  writer  was  J.  H.  Lambert  who  in  1766  wrote 
a  paper  "Zur  Theorie  der  Parallellinien,"  published  in  the  Leipziger 
Magazinfilr  reine  und  angewandte  Mathcmatik,  1786,  in  which:  (i)  The 
failure  of  the  parallel-axiom  in  surface-spherics  gives  a  geometry  with 
angle-sum  >2  right  angles;  (2)  In  order  to  make  intuitive  a  geometry 
with  angle-sum  <  2  right  angles  we  need  the  aid  of  an  "imaginary 
sphere"  (pseudo-sphere);  (3)  In  a  space  with  the  angle-sum  differing 
from  2  right  angles,  there  is  an  absolute  measure  (Bolyai's  natural 
unit  for  length).  Lambert  arrived  at  no  definite  conclusion  on  the 
validity  of  the  hypotheses  of  the  obtuse  and  acute  angles. 

Among  the  contemporaries  and  pupils  of  K.  F.  Gauss,  three  deserve 
mention  as  writers  on  the  theory  of  parallels,  Ferdinand  Karl  Schwei- 
kart  (i 780-1859),  professor  of  law  in  Marburg,  Franz  Adolf  Taurinus 
(i 794-1874),  a  nephew  of  Schweikart,  and  Friedrich  Ltidwig  Wackier 
(i792-i8i7),a  pupil  of  Gauss  in  1 809  and  professor  at  Dantzig.  Schwei- 
kart sent  Gauss  in  1818  a  manuscript  on  "Astral  Geometry"  which 
he  never  published,  in  which  the  angle-sum  of  a  triangle  is  less  than 
two  right  angles  and  there  is  an  absolute  unit  of  length.  He  induced 
Taurinus  to  study  this  subject.  Taurinus  published  in  1825  his  Theorie 
der  Parallellinien,  in  which  he  took  the  position  of  Saccheri  and  Lam- 
bert, and  in  1826  his  Geometrice  prima  elcmcnta,  in  an  appendix  of 
which  he  gives  important  trigonometrical  formulae  for  non-Euclidean 
geometry  by  using  the  formulae  of  spherical  geometry  with  an  imag- 
inary radius.  His  Elementa  attracted  no  attention.  In  disgust  he 
burned  the  remainder  of  his  edition.  Wachtcr's  results  are  contained 
in  a  letter  of  181 6  to  Gauss  and  in  his  Demonstratio  axiomatis  geo- 
metrici  in  Eiiclideis  undecimi,  181 7.  He  showed  that  the  geometry 
on  a  sphere  becomes  identical  with  the  geometry  of  Euclid  when  the 
radius  is  infinitely  increased,  though  it  is  distinctly  shown  that  the 
limiting  surface  is  not  a  plane.^    Elsewhere  we  have  mentioned  the 

1  See  English  translation  by  G.  B.  Halsted  in  Am.  Math.  Monthly,  Vols.  1-5,  1894- 
1898. 

^D.  M.  Y.  Sommerville,  Ekm.  of  N on- Euclidean  Geomdry,  London,  1914,  p.  15. 


3o6  A  HISTORY  OF  MATHEMATICS 

contemporary  researches  on  parallel  lines  due  to  A.  M.  Legendre  in 
France. 

The  researches  of  K.  F.  Gauss,  N.  I.  Lobachevski  and  J.  Bolyai 
have  been  considered  by  F.  Klein  as  constituting  the  first  period  in 
the  history  of  non-Euclidean  geometry.  It  is  a  period  in  which  the 
synthetic  methods  of  elementary  geometry  were  in  vogue.  The 
second  period  embraces  the  researches  of  G.  F.  B.  Riemann,  H.  Helm- 
holtz,  S.  Lie  and  E.  Beltrami,  and  employs  the  methods  of  differential 
geometry.  It  was  in  1854  that  Gauss  heard  from  his  pupil,  Riemann, 
a  marvellous  dissertation  which  considered  the  foundations  of  geome- 
try from  a  new  point  of  view.  Riemann  was  not  familiar  with  Lo- 
bachevski and  Bolyai.  He  developed  the  notion  of  n-ply  extended 
magnitude,  and  the  measure-relations  of  which  a  manifoldness  of  n 
dimensions  is  capable,  on  the  assumption  that  every  line  may  be 
measured  by  every  other.  Riemann  applied  his  ideas  to  space.  He 
taught  us  to  distinguish  between  " unboundedness "  and  "infinite 
extent."  According  to  him  we  have  in  our  mind  a  more  general  notion 
of  space,  i.  e.  a  notion  of  non-Euclidean  space;  but  we  learn  hy  expe- 
rience that  our  physical  space  is,  if  not  exactly,  at  least  to  a  high  degree 
of  approximation,  Euclidean  space.  Riemann's  profound  dissertation 
was  not  published  until  1S67,  when  it  appeared  in  the  Gottingen  Ab- 
handlungen.  Before  this  the  idea  of  n  dimensions  had  suggested  itself 
under  various  aspects  to  Ptolemy,  J.  Wallis,  D'Alembert,  J.  Lagrange, 
J.  Pliicker,  and  H.  G.  Grassmann.  The  idea  of  time  as  a  fourth  di- 
mension had  occurred  to  D'Alembert  and  Lagrange.  About  the  same 
time  with  Riemann's  paper,  others  were  published  from  the  pens  of 
H.  HelmhoUz  and  E.  Beltrami.  This  period  marks  the  beginning 
of  lively  discussions  upon  this  subject.  Some  writers — J.  Bellavitis, 
for  example — were  able  to  see  in  non-Euclidean  geometry  and  11- 
dimensional  space  nothing  but  huge  caricatures,  or  diseased  out- 
growths of  mathematics.  H.  Helmholtz's  article  was  entitled  That- 
sachen,  welche  der  Geomctrie  zu  Gninde  liegen,  1868,  and  contained 
many  of  the  ideas  of  Riemann.  Helmholtz  popularized  the  subject  in 
lectures,  and  in  articles  for  various  magazines.  Starting  with  the 
idea  of  congruence,  and  assuming  the  free  mobility  of  a  rigid  body  and 
the  return  unchanged  to  its  original  position  after  rotation  about  an 
axis,  he  proves  that  the  square  of  the  line-element  is  a  homogeneous 
function  of  the  second  degree  in  the  differentials.^  Helmholtz's 
investigations  were  carefully  examined  by  S.  Lie  who  reduced  the 
Riemann-Helmholtz  problem  to  the  following  form:  To  determine 
all  the  continuous  groups  in  space  which,  in  a  bounded  region,  have 
the  property  of  displacements.     There  arose  three  types  of  groups, 

Sommerville  is  the  author  of  a  Bibliography  of  non-Euclidean  geometry  including 
the  theory  of  parallels,  the  foiitidations  of  geometry,  atid  space  of  n  dimensions,  London, 
1911. 
1  D.  M.  Y.  Sommerville,  op.  cii.,  p.  195. 


SYNTHETIC  GEOMETRY 


30: 


which  characterize  the  three  geometries  of  Euclid,  of  N.  I.  Lobachev- 
ski  and  J.  Bolyai,  and  of  F.  G.  B.  Riemann.^ 

Eugenio  Beltrami  (1835- 1900),  born  at  Cremona,  was  a  pupil  of 
F.  JBrioschi.  He  was  professor  at  Bologna  as  a  colleague  of  L.  Cremona, 
at  Pisa  as  an  associate  of  E.  Betti,  at  Pavia  as  a  co-worker  with  F. 
Casorati,  and  since  iSgi  at  Rome  where  he  spent  the  last  years  of 
his  career,  "uno  degli  illustri  maestri  dell'  analisi  in  Italia."  Beltrami 
wrote  in  186S  a  classical  paper,  Saggio  di  interpretazione  delta  geometria 
non-euclidea  {Giorn.  di  Mateni.,  6),  which  is  analytical  (and,  like 
several  other  papers,  should  be  mentioned  elsewhere  were  we  to  adhere 
to  a  strict  separation  between  s^Tithesis  and  analysis).  He  reached 
the  brilliant  and  surprising  conclusion  that  in  part  the  theorems  of 
non-Euclidean  geometry  find  their  realization  upon  surfaces  of  con- 
stant negative  curvature.  He  studied,  also,  surfaces  of  constant 
positive  curvature,  and  ended  with  the  interesting  theorem  that 
the  space  of  constant  positive  curvature  is  contained  in  the  space 
of  constant  negative  curvature.  These  researches  of  Beltrami,  H. 
Helmholtz,  and  G.  F.  B.  Riemann  culminated  in  the  conclusion  that 
on  surfaces  of  constant  curvature  we  may  have  three  geometries, — 
the  non-Euclidean  on  a  surface  of  constant  negative  curvature, 
the  spherical  on  a  surface  of  constant  positive  curvature,  and  the 
Euclidean  geometry  on  a  surface  of  zero  curvature.  The  three  ge- 
ometries do  not  contradict  each  other,  but  are  members  of  a  system, — 
a  geometrical  trinity.  The  ideas  of  hyper-space  were  brilliantly  ex- 
pounded and  i:>opularised  in  England  by  Clifford. 

William  Kingdon  Clifford  (i 845-1 879)  was  born  at  Exeter,  edu- 
cated at  Trinity  College,  Cambridge,  and  from  187 1  until  his  death 
professor  of  applied  mathematics  in  University  College,  London.  His 
premature  death  left  incomplete  several  brilliant  researches  which 
he  had  entered  upon.  Among  these  are  his  paper  On  Classification 
of  Loci  and  his  Theory  of  Graphs.  He  wrote  articles  On  the  Canonical 
Form  and  Dissection  of  a  Riemann's  Surface,  on  Bi quaternions,  and 
ail  incomplete  work  on  the  Elements  of  Dynamic.  He  gave  exact 
nicaning  in  dynamics  to  such  familiar  w'ords  as  ''spin,"  "twist," 
"squirt,"  "whirl."  The  theory  of  polars  of  curves  and  surfaces  was 
generalized  by  him  and  by  Reye.  His  classification  of  loci,  1878, 
being  a  general  study  of  curves,  was  an  introduction  to  the  study 
of  ^-dimensional  space  in  a  direction  mainly  projective.  This  study 
has  been  continued  since  chiefly  by  G.  Veronese  of  Padua,  C.  Segre 
of  Turin,  E.  Bertini,  F.  Aschieri,  P.  Del  Pezzo  of  Naples. 

Beltrami's  researches  on  non-Euclidean  geometry  w^ere  followed, 
in  1 87 1,  by  important  investigations  of  Felix  Klein,  resting  upon 
Cayley's  Sixth  Memoir  on  Qualities,  1859.  The  development  of  geom- 
etry in  the  first  half  of  the  nineteenth  centur\^  had  led  to  the  separation 

'Lie,  Theorie  der  Transfonnationsgruppen,  Bd.  Ill,  Leipzig,  1893,  pp.  437-543; 
Bonola,  op.  cit.,  p.  154. 


3o8  A  HISTORY  OF  MA'i HEMATICS 

of  this  science  into  two  parts:  the  geometry  of  position  or  descr/ptive 
geometry  which  dealt  with  properties  that  are  unaffected  by  projec- 
tion, and  the  geometry  of  measurement  in  which  the  fundamental 
notions  of  distance,  angle,  etc.,  are  changed  by  projection.  Cayley's 
Sixth  Memoir  brought  these  strictly  segregated  parts  together  again 
by  his  definition  of  distance  between  two  points.  The  question 
whether  it  is  not  possible  so  to  express  the  metrical  properties  of 
figures  that  they  will  not  vary  by  projection  (or  linear  transformation) 
had  been  solved  for  special  projections  by  M.  Chasles,  J.  V.  Poncelet, 
and  E.  Laguerre,  but  it  remained  for  A.  Cayley  to  give  a  general 
solution  by  defining  the  distance  between  two  points  as  an  arbitrary 
constant  multiplied  by  the  logarithm  of  the  anharmonic  ratio  in  which 
the  line  joining  the  two  points  is  divided  by  the  fundamental  quadric. 
These  researches,  applying  the  principles  of  pure  projective  geometry, 
mark  the  third  period  in  the  development  of  non-Euclidean  geometry. 
Enlarging  upon  this  notion,  F.  Klein  showed  the  independence  of 
projective  geometry  from  the  parallel-axiom,  and  by  properly  choosing 
the  law  of  the  measurement  of  distance  deduced  from  projective 
geometry  the  spherical,  Euclidean,  and  pseudospherical  geometries, 
named  by  him  respectively  the  elliptic,  parabolic,  and  hyperbolic 
geometries.  This  suggestive  investigation  was  followed  up  by  numer- 
ous writers,  particularly  by  G.  Battaglini  of  Naples,  E.  d'Ovidio  of 
Turin,  R.  de  Paolis  of  Pisa,  F.  Aschieri,  A.  Cayley,  F.  Lindemann  of 
Munich,  E.  Schering  of  Gottingen,  W.  Story  of  Clark  University, 
H.  Stahl  of  Tiibingen,  A.  Voss  of  Munich,  Homersham  Cox,  A.  Buch- 
heim.^  The  notion  of  parallelism  applicable  to  hyperbolic  space  was 
the  only  extension  of  Euclid's  notion  of  parallelism  until  Clifford  dis- 
covered in  elliptic  space  straight  lines  which  possess  most  of  the  prop- 
erties of  Euclidean  parallels,  but  differ  from  them  in  being  skew.  Two 
Hues  are  right  (or  left)  parallel,  if  they  cut  the  same  right  (or  left) 
generators  of  the  absolute.  Later  F.  Klein  and  R.  S.  Ball  made 
extensive  contributions  to  the  knowledge  of  these  lines.  More  re- 
cently E.  Study  of  Bonn,  J.  L.  Coolidge  of  Harvard  University,  W. 
Vogt  of  Heidelberg  and  others  have  been  studying  this  subject.  The 
methods  employed  have  been  those  of  analytic  and  synthetic  geometry 
as  well  as  those  of  differential  geometry  and  vectorial  analysis.^  The 
geometry  of  n  dimensions  was  studied  along  a  line  mainly  metrical 
by  a  host  of  writers,  among  whom  may  be  mentioned  Simon  Newcomb 
of  the  Johns  Hopkins  University,  L.  Schlafii  of  Bern,  W.  I.  Stringham 
(1847-1909)  of  the  University  of  California,  W.  Killing  of  Miinster, 
T.  Craig  of  the  Johns  Hopkins,  Rudolf  Lipschitz  (i S3  2-1 903)  of  Bonn. 
R.  S.  Heath  of  Birmingham  and  W.  Killing  investigated  the  kinematics 
and  mechanics  of  such  a  space.  Regular  solids  in  /^-dimensional  space 
were  studied  by  Stringham,  Ellery  W.  Davis   (1857-IQ18)   of  the 

^  G.  Loria,  Die  hauptsdchlichslen  Theorien  der  Geometrie,  1S8S,  p.  102. 
^  Bull.  Am.  Math.  Soc,  Vol.  17,  1911,  p.  315. 


ANALYTIC  GEOMETRY  309 

University  of  Nebraska,  Rcinhold  Hoppe  (1816-1900^  of  Berlin,  and 
others.  Stringham  gave  pictures  of  projections  upon  our  space  of 
regular  solids  in  four  dimensions,  and  V.  Schlegel  at  Hagen  constructed 
models  of  such  projections.  These  are  among  the  most  curious  of  a 
series  of  models  published  by  L.  Brill  in  Darmstadt.  It  has  been 
pointed  out  that  if  a  fourth  dimension  existea,  certain  motions  could 
take  place  which  we  hold  to  be  impossible.  Thus  S.  Newcomb  showed 
the  possibility  of  turning  a  closed  material  shell  inside  out  by  simple 
flexure  without  either  stretching  or  tearing;  F.  Klein  pointed  out  that 
knots  could  not  be  tied;  G.  Veronese  showed  that  a  body  could  be 
removed  from  a  closed  room  without  breaking  the  walls;  C.  S.  Peirce 
proved  that  a  body  in  four-fold  space  either  rotates  about  two  axes 
at  once,  or  cannot  rotate  without  losing  one  of  its  dimensions. 

A  fourth  period  in  the  history  of  non-Euclidean  geometry,  intro- 
duced by  the  researches  of  Moritz  Pasch,  Giuseppe  Peano,  Mario 
Fieri,  David  Hilbert,  Oswald  Veblen,  concerns  itself  with  the  logical 
grounding  of  geometry  (including  non-Euclidean  forms)  upon  sets 
of  axioms. 

The  geometry  of  hyperspace  was  exploited  by  spiritualists  and 
mediums,  of  whom  Henry  Slade  was  the  most  notorious.  He  con- 
verted to  spiritualism  the  German  scientist  F.  Zollner  and  his  coterie, 
to  whom  he  gave  a  spiritual  demonstration  of  the  existence  of  a  fourth 
dimension  of  space.  These  events  contributed  to  the  severity  with 
which  the  philosopher  R.  H.  Lotze,  in  his  Metaphysik,  1879,  criticised 
the  mathematical  theories  of  hyperspace  and  non-Euclidean  geometry. 

Analytic  Geometry 

In  the  preceding  chapter  we  endeavored  to  give  a  flashlight  view 
of  the  rapid  advance  of  synthetic  geometry.  In  some  cases  we  also 
mentioned  analytical  treatises.  Modern  synthetic  and  modern 
analytical  geometry  have  much  in  common,  and  may  be  grouped  to- 
gether under  the  common  name  "projective  geometry."  Each  has 
advantages  over  the  other.  The  continual  direct  viewing  of  figures 
as  existing  in  space  adds  exceptional  charm  to  the  study  of  the  former, 
but  the  latter  has  the  advantage  in  this,  that  a  well-established  routine 
in  a  certain  degree  may  outrun  thought  itself,  and  thereby  aid  original 
research.  While  in  Germany  J.  Steiner  and  von  Staudt  developed 
synthetic  geometry,  Pliicker  laid  the  foundation  of  modern  analytic 
geometry. 

Julius  Pliicker  (1801-1868)  was  born  at  Elberfeld,  in  Prussia.  After 
studying  at  Bonn,  Berlin,  and  Heidelberg,  he  spent  a  short  time  in 
Paris  attending  lectures  of  G.  Monge  and  his  pupils.  Between  1826 
and  1836  he  held  positions  successively  at  Bonn,  Berlin,  and  Halle. 
He  then  became  professor  of  physics  at  Bonn.  Until  1846  his  original 
researches  were  on  geometry.    In  1828  and  in  1831  he  published  hi« 


3IO  A  HISTORY  OF  MATHEMATICS 

Analytisch-Geometrtsche  Entwicklungen  in  two  volumes.  Therein  he 
adopted  the  abbreviated  notation  [used  before  him  in  a  more  restricted 
way  by  Etienne  Bobillier  (1797-1S32),  professor  of  mechanics  at 
Chalons-sur-Marne],  and  avoided  the  tedious  process  of  algebraic 
elimination  by  a  geometric  consideration.  In  the  second  volume  the 
principle  of  duality  is  formulated  analytically.  With  him  duality  and 
homogeneity  found  expression  already  in  his  system  of  co-ordinates. 
The  homogenous  or  trilinear  system  used  by  him  is  much  the  same  as 
the  co-ordinates  of  A.  F.  Mobius.  In  the  identity  of  analytical  opera- 
tion and  geometric  construction  Pliicker  looked  for  the  source  of  his 
proofs.  The  System  der  Analytischen  Gcometrie,  1S35,  contains  a 
complete  classification  of  plane  curves  of  the  third  order,  based  on  the 
nature  of  the  points  at  infinity.  The  Theorie  der  Algebraischen  Curven, 
1839,  contains,  besides  an  enumeration  of  curves  of  the  fourth  order, 
the  analytic  relations  between  the  ordinary  singularities  of  plane 
curves  known  as  "Pliicker's  equations,"  by  which  he  was  able  to 
explain  "Poncelet's  paradox."  The  discovery  of  these  relations  is, 
says  A.  Cayley,  "the  most  important  one  beyond  all  comparison  in 
the  entire  subject  of  modern  geometry."  The  four  Pliicker  equa- 
tions have  been  expressed  in  different  forms.  Cayley  studied  higher 
singularities  of  plane  curves.  M.  W.  Haskell  of  the  University  of 
California,  in  1914,  showed  from  the  Pliicker  equations  that  the 
maximum  number  of  cusps  possible  for  a  curve  of  order  m  is  the 
greatest  integer  in  m  (w— 2)73  (except  when  w  is  4  and  6,  in  which 
case  the  maximum  number  is  3  and  9),  and  that  there  is  always  a 
self-dual  curve  with  this  maximum  number  of  cusps. 

Certain  interrelations  of  the  various  geometrical  researches  of  the 
first  half  and  middle  of  the  nineteenth  century  are  brought  out  by 
J.  G.  Darboux  in  the  following  passage:^  "While  M.  Chasles,  J. 
Steiner,  and,  later,  .  .  .  von  Staudt,  were  intent  on  constituting  a 
rival  doctrine  to  analysis  and  set  in  some  sort  altar  against  altar, 
J.  D.  Gergonne,  E.  Bobilher,  C.  Sturm,  and  above  all  J.  Pliicker,  per- 
fected the  geometry  of  R.  Descartes  and  constituted  an  analytic  sys- 
tem in  a  manner  adequate  to  the  discoveries  of  the  geometers.  It  is 
to  E.  Bobillier  and  to  J.  Pliicker  that  we  owe  the  method  called 
abridged  notation.  Bobillier  consecrated  to  it  some  pages  truly  new 
in  the  last  volumes  of  the  Annates  of  Gergonne.  Pliicker  commenced 
to  develop  it  in  his  first  work,  soon  followed  by  a  series  of  works  where 
are  established  in  a  fully  conscious  manner  the  foundations  of  the 
modern  analytic  geometry.  It  is  to  him  that  we  owe  tangential  co- 
ordinates, trilinear  co-ordinates,  employed  with  homogeneous  equa- 
tions, and  finally  the  employment  of  canonical  forms  whose  validity 
was  recognized  by  the  method,  so  deceptive  sometimes,  but  so  fruit- 
ful, called  the  enumeration  of  constants. '^ 

In  Germany  J.  Pliicker's  researches  met  with  no  favor.  His  method 
'  Congress  of  Arts  and  Science,  St.  Louis,  1904,  Vol.  1,  pp.  541,  542. 


ANALYTIC  GEOMETRY  311 

was  declared  to  be  unproductive  as  compared  with  the  synthetic 
method  of  J.  Steiner  and  J.  V.  Poncelet!  His  relations  with  C.  G.  J. 
Jacobi  were  not  altogether  friendly.  Steiner  once  declared  that  he 
would  stop  writing  for  Crelle's  Journal  if  Pliicker  continued  to  con- 
tribute to  it.^  The  result  was  that  many  of  Pliicker's  researches  were 
published  in  foreign  journals,  and  that  his  work  came  to  be  better 
known  in  France  and  England  than  in  his  native  country.  The  charge 
was  also  brought  against  Pliicker  that,  though  occupying  the  chair 
of  physics,  he  was  no  physicist.  This  induced  him  to  relinquish 
mathematics,  and  for  nearly  twenty  years  to  devote  his  energies  to 
physics.  Important  discoveries  on  Fresnel's  wave-surface,  magnetism, 
spectrum-analysis  were  made  by  him.  But  towards  the  close  of  his 
life  he  returned  to  his  first  love, — mathematics, — and  enriched  it  with 
new  discoveries.  By  considering  space  as  made  up  of  lines  he  created 
a  "new  geometry  of  space."  Regarding  a  right  line  as  a  curve  in- 
volving four  arbitrary  parameters,  one  has  the  whole  system  of  lines 
in  space.  By  connecting  them  by  a  single  relation,  he  got  a  "  complex  " 
of  lines;  by  connecting  them  with  a  twofold  relation,  he  got  a  "con- 
gruency"  of  lines.  His  first  researches  on  this  subject  were  laid  before 
the  Royal  Society  in  1865.  His  further  investigations  thereon  ap- 
peared in  1868  in  a  posthumous  work  entitled  Neue  Geometrie  des 
Raumes  gegriindet  auf  die  Betrachtung  der  geraden  Linie  als  Raumele- 
ment,  edited  by  Felix  Klein.  Pliicker's  analysis  lacks  the  elegance 
found  in  J.  Lagrange,  C.  G.  J.  Jacobi,  L.  O.  Hesse,  and  R.  F.  A. 
Clebsch.  For  many  years  he  had  not  kept  up  with  the  progress  of 
geometry,  so  that  many  investigations  in  his  last  work  had  already 
received  more  general  treatment  on  the  part  of  others.  The  work 
contained,  nevertheless,  much  that  was  fresh  and  original.  The  theory 
of  complexes  of  the  second  degree,  left  unfinished  by  Pliicker,  was 
continued  by  Felix  Klein,  who  greatly  extended  and  supplemented 
the  ideas  of  his  master. 

Ludwig  Otto  Hesse  (1811-1S74)  was  born  at  Konigsberg,  and 
studied  at  the  university  of  his  native  place  under  F.  W.  Bessel,  C.  G.  J. 
Jacobi,  F.  J.  Richelot,  and  F.  Neumann.  Having  taken  the  doctor's 
degree  in  1S40,  he  became  docent  at  Konigsberg,  and  in  1845  extraor- 
dinary professor  there.  Among  his  pupils  at  that  time  were  Heinrich 
Diirege  (1821-1893)  of  Prague,  Carl  Neumann,  R.  F.  A.  Clebsch, 
G.  R.  KirchhofT.  The  Konigsberg  period  was  one  of  great  activity 
for  Hesse.  Every  new  discovery  increased  his  zeal  for  still  greater 
achievement.  His  earliest  researches  were  on  surfaces  of  the  second 
order,  and  were  partly  synthetic.  He  solved  the  problem  to  construct 
any  tenth  point  of  such  a  surface  when  nine  points  are  given.  The 
analogous  problem  for  a  conic  had  been  solved  by  Pascal  by  means 
of  the  hexagram.  A  difficult  prol:)lem  confronting  mathematicians 
of  this  time  was  that  of  elimination.  J.  Pliicker  had  seen  that  the 
1  Ad.  Dronke,  Julius  Pliicker,  Bonn,  1871. 


312  A  HISTORY  OF  MATHEMATICS 

main  advantage  of  his  special  method  in  analytic  geometry  lay  m 
the  avoidance  of  algebraic  elimination.  Hesse,  however,  showed  how 
by  determinants  to  make  algebraic  elimination  easy.  In  his  earlier 
results  he  was  anticipated  by  J.  J.  Sylvester,  who  published  his  dialytic 
method  of  elimination  in  1840.  These  advances  in  algebra  Hesse 
applied  to  the  analytic  study  of  curves  of  the  third  order.  By  linear 
substitutions,  he  reduced  a  form  of  the  third  degree  in  three  variables 
to  one  of  only  four  terms,  and  was  led  to  an  important  determinant 
involving  the  second  differential  coefficient  of  a  fonn  of  the  third 
degree,  called  the  "Hessian."  The  "Hessian"  plays  a  leading  part 
in  the  theory  of  invariants,  a  subject  first  studied  by  A.  Cayley. 
Hesse  showed  that  his  determinant  gives  for  every  curve  another 
curve,  such  that  the  double  points  of  the  first  are  points  on  the  second, 
or  "Hessian."  Similarly  for  surfaces  (Crelle,  1S44).  Many  of  the 
most  important  theorems  on  curves  of  the  third  order  are  due  to 
Hesse.  He  determined  the  curve  of  the  14th  order,  which  passes 
through  the  56  points  of  contact  of  the  28  bi-tangents  of  a  curve  of 
the  fourth  order.  His  great  memoir  on  this  subject  (Crelle,  1855) 
was  published  at  the  same  time  as  was  a  paper  by  J.  Steiner  treating 
of  the  same  subject. 

Hesse's  income  at  Konigsberg  had  not  kept  pace  with  his  growing 
reputation.  Hardly  w^as  he  able  to  support  himself  and  family.  In 
1855  he  accepted  a  more  lucrative  position  at  Halle,  and  in  1856  one 
at  Heidelberg.  Here  he  remained  until  1S68,  when  he  accepted  a 
position  at  a  technic  school  in  Munich.^  At  Heidelberg  he  revised 
and  enlarged  upon  his  previous  researches,  and  published  in  1861  his 
Vorlesungen  uber  die  Analytische  Geomeirie  des  Raiimes,  insbesondere 
ilber  Fl'dchen.  2.  Ordnung.  More  elementary  works  soon  followed. 
While  in  Heidelberg  he  elaborated  a  principle,  his  "  Uebertragungs- 
princip."  According  to  this,  there  corresponds  to  every  point  in  a 
plane  a  pair  of  points  in  a  line,  and  the  projective  geometry  of  the 
plane  can  be  carried  back  to  the  geometry  of  points  in  a  hne. 

The  researches  of  Pliicker  and  Hesse  were  continued  in  England 
by  A.  Cayley,  G.  Salmon,  and  J.  J.  Sylvester.  It  may  be  premised 
here  that  among  the  early  writers  on  analytical  geometry  in  England 
was  James  Booth  (1S06-1878),  whose  chief  results  are  embodied  in  his 
Treatise  on  Some  New  Geometrical  Methods;  and  James  MacCullagh 
(1809-1846),  who  was  professor  of  natural  philosophy  at  Dublin, 
and  made  some  valuable  discoveries  on  the  theory  of  quadrics.  The 
influence  of  these  men  on  the  progress  of  geometry  was  insignificant, 
for  the  interchange  of  scientific  results  between  different  nations  was 
not  so  complete  at  that  time  as  might  have  been  desired.  In  further 
illustration  of  this,  we  mention  that  M.  Chasles  in  France  elaborated 
subjects  which  had  previously  been  disposed  of  by  J.  Steiner  in  Ger- 
many, and  Steiner  published  researches  which  had  been  given  by 
'-  Gustav  Bauer,  Ged'dchlnissrede  an f  Otto  Hesse.  Miinchen,  1S82. 


ANALYTIC  GEOMETRY 


3^3 


Cayley,  Sylvester,  and  Salmon  nearly  five  years  earlier.  Cayley  and 
Salmon  in  1S49  determined  the  straight  lines  in  a  cubic  surface,  and 
studied  its  principal  properties,  while  Sylvester  in  185 1  discovered 
the  pentahedron  of  such  a  surface.  Cayley  extended  Pliicker's  equa- 
tions to  curves  of  higher  singularities.  Cayley's  owni  investigations, 
and  ihose  of  Max  Nother  (1844-  )  of  Erlangen,  G.  H.  Halphen, 
Jules  R.  M.  de  la  Gournerie  (1S14-18S3)  of  Paris,  A.  Brill  of  Tubin- 
gen, lead  to  the  conclusion  that  each  higher  singularity  of  a  curve  is 
equivalent  to  a  certain  number  of  simple  singularities, — the  node,  the 
ordinary  cusp,  the  double  tangent,  and  the  inflection.  Syh-ester 
studied  the  "  twisted  Cartesian,"  a  curve  of  the  fourth  order.  Georges- 
Henri  Halphen  (1844-1SS9)  was  born  at  Rouen,  studied  at  the  Ecole 
Polytechnique  in  Paris,  took  part  in  the  Franco-Prussian  war,  then 
became  repetiteur  and  examinateur  at  the  Ecole  Polytechnique.  His 
investigations  touched  mainly  the  geometry  of  algebraic  curv'es  and 
surfaces,  differential  invariants,  the  theory  of  E.  Laguerre's  invariants, 
elliptic  functions  and  their  applications.  A  British  geometrician, 
Salmon,  helped  powerfully  towards  the  spreading  of  a  knowledge  of 
the  new  algebraic  and  geometric  methods  by  the  publication  of  an 
excellent  series  of  text-books  (Conic  Sections,  Modern  Higher  Algebra, 
Higher  Plane  Curves,  Analytic  Geometry  of  Three  Dimensions),  which 
have  been  placed  within  easy  reach  of  German  readers  by  a  free  trans- 
lation, with  additions,  made  by  Wilhelm  Fiedler  of  the  Polytechnicum 
in  Zurich.  Salmon's  Geometry  of  Three  Dimensions  was  brought  out 
in  the  fifth  and  sixth  editions,  with  much  new  matter,  by  Reginald 
A.  P.  Rogers  of  Trinity  College,  Dublin,  in  1912-1915.  The  next  great 
worker  in  the  field  of  analytic  geometry  was  Clebsch. 

Rudolf  Friedrich  Alfred  Clebsch  (1833-1872)  was  born  at  Konigs- 
berg  in  Prussia,  studied  at  the  university  of  that  place  under  L.  0. 
Hesse,  F.  J.  Richclot,  F.  Neumann.  From  1858  to  1863  he  held  the 
chair  of  theoretical  mechanics  at  the  Polytechnicum  in  Carlsruhe. 
The  study  of  Salmon's  works  led  him  into  algebra  and  geometry.  In 
1863  he  accepted  a  position  at  the  University  of  Giesen,  where  he 
worked  in  conjunction  with  Paul  Gordan  of  Erlangen.  In  1868 
Clebsch  went  to  Gottingen,  and  remained  there  until  his  death.  He 
worked  successively  at  the  following  subjects:  Mathematical  physics, 
the  calculus  of  variations  and  partial  differential  equations  of  the  first 
order,  the  general  theory  of  curves  and  surfaces,  Abelian  functions 
and  their  use  in  geometry,  the  theory  of  invariants,  and  "Flachen- 
abbildang."  ^  He  proved  theorems  on  the  pentahedron  enunciated 
by  J.  J.  Sylvester  and  J.  Steiner;  he  made  systematic  use  of  "defi- 
ciency" (Geschlecht)  as  a  fundamental  principle  in  the  classification 
of  algebraic  curves.  The  notion  of  deficiency  was  known  before  him 
to  N.  H.  Abel  and  G.  F.  B.  Riemann.    At  the  beginning  of  his  career, 

'  .Alfred  Clebsch,  Vcrsuch  ei'icr  Darlegung  uud  Wiirdigung  seiner  uissrnsrkafllichen 
"'.t'itiini^cii  von  einigcn  seiner  Freunde,  Leipzig,  1873. 


314  A  HISTORY  OF  MATHEMATICS 

Clebsch  had  shown  how  elliptic  functions  could  be  advantageously 
applied  to  Malfatti's  problem.  The  idea  involved  therein,  viz.  the  use 
of  higher  transcendentals  in  the  study  of  geometry,  led  him  to  his  great- 
est discoveries.  Not  only  did  he  apply  Abelian  functions  to  geometry, 
but  conversely,  he  drew  geometry  into  the  service  of  Abelian  functions. 

Clebsch  made  liberal  use  of  determinants.  His  study  of  curves  and 
surfaces  began  with  the  determination  of  the  points  of  contact  of  lines 
which  meet  a  surface  in  four  consecutive  points.  G.  Salmon  had 
proved  that  these  points  lie  on  the  intersection  of  the  surface  with  a 
derived  surface  of  the  degree  ii;^  — 24,  but  his  solution  was  given  in 
inconvenient  form.  Clebsch's  investigation  thereon  is  a  most  beautiful 
piece  of  analysis. 

The  representation  of  one  surface  upon  another  {Flachenahhildung) , 
so  that  they  have  a  (i,  i)  correspondence,  was  thoroughly  studied  for 
the  first  time  by  Clebsch.  The  representation  of  a  sphere  on  a  plane 
is  an  old  problem  which  drew  the  attention  of  Ptolemy,  Gerard  Mer- 
cator,  J.  H.  Lambert,  K.  F.  Gauss,  J.  L.  Lagrange.  Its  importance  in 
the  construction  of  maps  is  obvious.  Gauss  was  the  first  to  represent 
a  surface  upon  another  with  a  view  of  more  easily  arriving  at  its 
properties.  J.  Plucker,  M.  Chasles,  A.  Cayley,  thus  represented  on  a 
plane  the  geometry  of  quadric  surfaces;  Clebsch  and  L.  Cremona,  that 
of  cubic  surfaces.  Other  surfaces  have  been  studied  in  the  same  way 
by  recent  writers,  particularly  Max  N other  of  Erlangen,  Angelo 
Arnienante  (1S44-1878)  of  Rome,  Felix  Klein,  Georg  II.  L.  Korndorjcr, 
Ettore  Caporali  (1855-1S86)  of  Naples,  //.  G.  Zeuthen  of  Copenhagen, 
A  fundamental  question  which  has  as  yet  received  only  a  partial  an- 
swer is  this:  What  surfaces  can  be  represented  by  a  (i,  i)  correspond- 
ence upon  a  given  surface?  This  and  the  analogous  question  for 
curves  was  studied  by  Clebsch.  Higher  correspondences  between 
surfaces  have  been  investigated  by  A.  Cayley  and  M.  Nother.  Im- 
portant bearings  upon  geometry  has  Riemann's  theory  of  birational 
transformations.  The  theory  of  surfaces  has  been  studied  by  Joseph. 
Alfred  Serret  (1819-1885)  professor  at  the  Sorbonne  in  Paris,  Jean 
Gaston  Darboux  of  Paris,  John  Casey  (1820-1891)  of  Dublin,  William 
Roberts  (1817-1883)  of  Dublin,  Heinrich  Schrotcr  (1829-1S92)  of 
Breslau,  Elwin  Bruno  Christoffel  (1829-1900),  professor  at  Zurich, 
later  at  Strassburg.  Christoffel  WTote  on  the  theory  of  potential,  on 
minimal  surfaces,  on  the  so-called  transformation  of  Christoffel,  of 
isothermic  surfaces,  on  the  general  theory  of  cur^^ed  surfaces.  His 
researches  on  surfaces  were  extended  by  Julius  Weingartcn  (1836-1910) 
of  the  University  of  Freiburg  and  Ilans  von  Mangoklt  of  Aachen,  in 
1882.  As  we  shall  see  more  fully  later,  surfaces  of  the  fourth  order 
were  investigated  by  E.  E.  Kummer,  and  Fresnel's  wave-surface, 
studied  by  W.  R.  Hamilton,  is  a  particular  case  of  Rummer's  quartic 
surface,  with  sixteen  double  points  and  sixteen  singular  tangent  planes.'^- 
1  A.  Cayley,  Inaugural  Address,  1883. 


ANALYTIC  GEOMETRY  315 

Prominent  in  these  geometric  researches  was  Jean  Gaston  Dar- 

boux  (1842-1917).  He  was  born  at  Nimes,  founded  in  1870,  with  the 
collaboration  of  GuiUaunie  Jules  Iloilel  (1823-1886)  of  Bordeaux  and 
Jules  Tannery,  the  Bulletin  des  sciences  mathematiques  et  astronomiques, 
and  was  for  half  a  century  conspicuous  as  a  teacher.  In  1900  he 
became  permanent  secretary  of  the  Paris  Academy  of  Sciences,  in 
Avhich  position  he  was  succeeded  after  his  death  by  Emil  Picard.  By 
his  researches,  Darboux  enriched  the  synthetic,  analytic  and  infin- 
itesimal geometries,  as  well  as  rational  mechanics  and  analysis.  He 
^\Tote  Leqons  sur  la  theorie  gcncrale  des  surfaces  et  les  applications 
geomctriqiies  du  calcul  infinitesimal,  Paris,  1887-1896,  and  Leqons  sur 
les  systemes  orthogonaux  ct  les  coordonnees  curvilignes,  Paris,  1898.  He 
investigated  triply  orthogonal  systems  of  surfaces,  the  deformation 
of  surfaces  and  rolling  of  applicable  surfaces,  infinitesimal  deforma- 
tion, spherical  representation  of  surfaces,  the  development  of  the 
moving  axes  of  co-ordinates,  the  use  of  imaginary  geometric  elements, 
the  use  of  isotropic  cylinders  and  developables;  ^  he  introduced 
pentaspherical  coordinates. 

Eisenhart  says:  "Darboux  was  a  strong  advocate  of  the  use  of 
imaginary  elements  in  the  study  of  geometry.  He  beHeved  that  their 
use  was  as  necessary  in  geometry  as  in  analysis.  He  has  been  im- 
pressed by  the  success  with  which  they  have  been  employed  in  the 
solution  of  the  problem  of  minimal  surfaces.  From  the  very  beginning 
he  made  use  in  his  papers  of  the  isotropic  line,  the  null  sphere  (the 
isotropic  cone)  and  the  general  isotropic  developable.  In  his  first 
memoir  on  orthogonal  systems  of  surfaces  he  showed  that  the  envelope 
of  the  surfaces  of  such  a  system,  when  defined  by  a  single  equation, 
is  an  isotropic  developable.  .  .  .  Darboux  gives  to  Edouard  Com- 
bescure  (1819-?)  the  credit  of  being  the  first  to  apply  the  considera- 
tions of  kinematics  to  the  study  of  the  theory  of  surfaces  with  the 
consequent  use  of  moving  co-ordinate  axes.  But  to  Darboux  we  are 
indebted  for  a  realization  of  the  power  of  this  method,  and  for  its 
systematic  development  and  exposition.  .  .  .  Darboux's  ability  was 
based  on  a  rare  combination  of  geometrical  fancy  and  analytical 
power.  He  did  not  sympathize  with  those  who  use  only  geometrical 
reasoning  in  attacking  geometrical  problems,  nor  with  those  who  feel 
that  there  is  a  certain  virtue  in  adhering  strictly  to  analytical  proc- 
esses. ...  In  common  with  Monge  he  was  not  content  with  dis- 
coveries, but  felt  that  it  was  equally  important  to  make  disciples. 
Like  this  distinguished  ]5redecessor  he  developed  a  large  group  of 
geometers,  including  C.  Guichard,  G.  Koenigs,  E.  Cosserat,  A.  De- 
moulin,  G.  Tzitzeica,  and  G.  Demartrcs.  Their  brilHant  researches 
are  the  best  tribute  to  his  teaching." 

Proceeding  to  the  fuller  consideration  of  recent  developments,  we 

'  Am.  Math.  Mortlhly,  Vol.  24,  1017,  p.  354.  See  L.  P.  Eisenhart's  "Darboux's 
Contribution  to  Geometry"  in  Bull.  Am.  Math.  Soc,  Vol.  24,  1018,  p.  227. 


3i6  A  HISTORV  OF  MATHEMATICS 

quote  from  H.  F.  Baker's  address  before  the  International  Congress 
held  at  Cambridge  in  191 2:  ^  "The  general  theory  of  Higher  Plane 
Curves  .  .  .  would  be  impossible  without  the  notion  of  the  genus  of  a 
curve.  The  investigation  of  Abel  of  the  number  of  independent  in- 
tegrals in  terms  of  which  his  integral  sums  can  be  expressed  may  thus 
be  held  to  be  of  paramount  importance  for  the  general  theory.  This 
was  further  emphasized  by  G.  F.  B.  Riemann's  consideration  of  the  no- 
tion of  birational  transformation  as  a  fundamental  principle.  After 
this  two  streams  of  thought  were  to  be  seen.  First  R.  F.  A.  Clebsch 
remarked  on  the  existence  of  invariants  for  surfaces,  analogous  to  the 
genus  of  a  plane  curve.  This  number  he  defined  by  a  double  integral ; 
it  was  to  be  unaltered  by  birational  transformation  of  the  surface. 
Clebsch's  idea  was  carried  on  and  developed  by  M.  Noether.  But 
also  A.  Brill  and  Noether  elaborated  in  a  geometrical  form  the 
results  for  plane  curves  which  had  been  obtained  with  transcen- 
dental considerations  by  N.  H.  Abel  and  G.  F.  B.  Riemann.  Then 
the  geometers  of  Italy  took  up  Noether's  work  with  very  remark- 
able genius,  and  carried  it  to  a  high  pitch  of  perfection  and  clear- 
ness as  a  geometrical  theory.  In  connection  therewith  there  arose 
the  important  fact,  which  does  not  occur  in  Noether's  papers,  that 
it  is  necessary  to  consider  a  surface  as  possessing  two  genera;  and 
the  names  of  A.  Cayley  and  H.  G.  Zeuthen  should  be  referred  to  at 
this  stage.  But  at  this  time  another  stream  was  running  in  France. 
E.  Picard  was  developing  the  theory  of  Riemann  integrals — single 
integrals,  not  double  integrals — upon  a  surface.  How  long  and 
laborious  was  the  task  may  be  judged  from  the  fact  that  the  publica- 
tion of  Picard's  book  occupied  ten  years — and  may  even  then  have 
seemed  to  many  to  be  an  artificial  and  unproductive  imitation  of 
the  theory  of  algebraic  integrals  for  a  curve.  In  the  light  of  subse- 
quent events,  Picard's  book  appears  likely  to  remain  a  permanent 
landmark  in  the  history  of  geometry.  For  now  the  two  streams, 
the  purely  geometrical  in  Italy,  the  transcendental  in  France,  have 
united.  The  results  appear  to  me  at  least  to  be  of  the  greatest  im- 
portance." The  work  of  E.  Picard  in  question  is  the  Theorie  des 
fonctions  algehriques  de  deux  variables  indep  end  antes,  which  was  brought 
out  in  conjunction  with  Georges  Simart  between  the  years  1897  and 
1906. 

H.  F.  Baker  proceeds  to  the  enumeration  of  some  individual  re- 
sults: Guido  Castelnuovo  of  Rome  has  shown  that  the  deficiency  of 
the  characteristic  series  of  a  linear  system  of  curves  upon  a  surface  can- 
not exceed  the  difference  of  the  two  genera  of  the  surface.  Federigo 
Enriques  of  Bologna  has  completed  this  result  by  showing  that  for  an 
algebraic  system  of  curves  the  characteristic  series  is  complete.  Upon 
this  result,  and  upon  E.  Picard's  theory  of  integrals  of  the  second 

^Proceed,  of  the  5th  Inlern.  Congress,  Vol.  I,  Cambridge,  1913,  p.  49.  For  more 
detail,  consult  H.  B.  Baker  in  the  Proceed,  of  the  London  Math.  Soc,  Vol,  12,  1912. 


ANALYTIC  GEOMETRY  317 

kind  Francesco  Severi  of  Padua  has  constructed  a  proof  that  the 
number  of  Picard  integrals  of  the  first  kind  upon  a  surface  is  equal  to 
the  difference  of  the  genera.  The  names  of  M.  G.  Humbert  of  Paris 
and  of  G.  Castelnuovo  also  arise  here.  Picard's  theory  of  integrals  of 
the  third  kind  has  given  rise  in  F.  Severi's  hands  to  the  expression  d 
any  curve  lying  on  a  surface  linearly  in  terms  of  a  finite  number  of 
fundamental  curves.  Enriques  showed  that  the  system  of  curves  cut 
upon  a  plane  by  adjoint  surfaces  of  order  w— 3,  when  n  is  the  order  of 
the  fundamental  surface,  if  not  complete,  has  a  deficiency  not  ex- 
ceeding the  difference  of  the  genera  of  the  surface.  Severi  has  given 
a  geometrical  proof  that  this  deficiency  is  equal  to  the  difference  of 
the  genera,  a  result  previously  deduced  by  E.  Picard,  with  transcen- 
dental considerations,  from  the  assumption  of  the  number  of  Picard 
integrals  of  the  first  kind.  F.  Enriques  and  G.  Castlenuovo  have  shown 
that  a  surface  which  possesses  a  system  of  curves  for  which  what  may 
be  called  the  canonical  number,  2T—  2  —  n,  where  7r  is  the  genus  of  the 
curve  and  n  the  number  of  intersections  of  two  curves  of  the  system, 
is  negative,  can  be  transformed  birationally  to  a  ruled  surface.  On 
the  analog}'  of  the  case  of  plane  curves,  and  of  surfaces  in  three  di- 
mensions, it  appears  very  natural  to  conclude  that  if  a  rational  re- 
lation, connecting,  say,  m+i  variables,  can  be  resolved  by  substi- 
tuting for  the  variables  rational  functions  of  m  others,  then  these  m 
others  can  be  so  chosen  as  to  be  rational  functions  of  the  m+i  original 
variables.  F.  Enriques  has  recently  given  a  case,  with  ^=3,  for 
which  this  is  not  so.  To  this  summary  of  results,  given  by  H.  F. 
Baker,  should  be  added  that  he  himself  has  made  contributions, 
particularly  on  a  cubic  surface  and  the  curves  which  lie  thereon. 
In  reducing  singularities,  the  Italians  and  French  use  methods  of 
projecting  from  space  of  higher  dimension  which  were  perhaps  first 
used  in  1SS7  by  W.  K.  Clifford. 

A  publication  of  wide  scope  on  collineations  and  correlations  is 
Die  Lehre  von  den  geomelrischen  Verwandtschaften,  in  four  volumes, 
1908 — ?,  being  written  by  Rudolf  Sturm  (1S41 — )  of  the  University 
of  Breslau. 

The  theory  of  straight  lines  upon  a  cubic  surface  was  first  studied 
by  A.  Cayley  and  G.  Salmon  ^  in  1849.  Cayley  pointed  out  that  there 
was  a  definite  number  of  such  lines,  while  Salmon  found  that  there 
were  exactly  27  of  them.  "Surely  with  as  good  reason,"  says  J.  J. 
Sylvester,  "as  had  Archimedes  to  have  the  cylinder,  cone  and  sphere 
engraved  on  his  tombstone  might  our  distinguished  countr\Tnen  leave 
testamentary  directions  for  the  cubic  eikosiheptagram  to  be  engraved 
on  theirs."  Nor  would  such  engraving  be  impossible,  for  in  1869 
Christian  Wiener  made  a  model  of  a  cubic  surface  showing  27  real 
lines  lying  upon  it.    J.  Steincr,  in  1856,  studied  the  purely  geometric 

'  These  historic  data  are  taken  from  A.  Henderson,  The  Twaity-sevcn  Lines  upon 
he  Cubic  Surface,  Cambridge,  1911,  which  give=  bibliography  and  details. 


3i8  A  HISTORY  OF  MATHEMATICS 

theory  of  cubic  surfaces.  This  was  done  later  also  by  L.  Cremona  and 
R.  Sturm,  between  whom  in  1866  the  "Steiner  prize"  was  divided. 
An  elegant  notation  was  invented  by  Andrew  Hart,  but  the  notation 
which  has  met  with  general  adoption  was  advanced  by  L.  Schlafli  of 
Bern  in  1858;  it  is  that  of  the  double  six.  Schlafii's  double  six  theorem, 
proved  by  him  and  by  many  others  since,  is  as  follows:  "Given  five 
lines  a,  b,  c,  d,  e  which  meet  the  same  straight  line  X;  then  may  any 
four  of  the  five  lines  be  intersected  by  another  line.  Suppose  that 
A,  B,  C,  D,  E  are  the  other  lines  intersecting  {b,  c,  d,  e),  (c,  d,  e,  a), 
{d,  e,  a,  b),  (e,  a,  b,  c),  and  (a,  b,  c,  d)  respectively.  Then  A,  B,  C,  D,  E 
will  all  be  met  by  one  other  straight  line  x." 

L.  Schlafli  first  considered  a  division  of  the  cubic  surfaces  into 
species,  in  regard  to  the  reality  of  the  27  lines.  His  final  classification 
was  adopted  by  A.  Cayley.  In  1S72  R.  F.  A.  Clebsch  constructed  a 
model  of  the  diagonal  surface  with  27  real  lines,  while  F.  Klein  "es- 
tablished the  fact  that  by  the  principle  of  continuity  all  forms  of  real 
surfaces  of  the  third  order  can  be  derived  from  the  particular  surface 
having  four  conical  points;"  he  exhibited  a  complete  set  of  models 
of  cubic  surfaces  at  the  World's  Fair  in  Chicago  in  1894.  In  1869 
C.  F.  Geiser  showed  that "  the  projection  of  a  cubic  surface  from  a  point 
upon  it,  on  a  plane  of  projection  parallel  to  the  tangent  plane  at  that 
point,  is  a  quartic  cur\^e;  and  that  every  quartic  curve  can  be  generated 
in  this  way."  "The  theory  of  varieties  of  the  third  order,"  says  A. 
Henderson,  "that  is  to  say,  curved  geometric  forms  of  three  dimen- 
sions contained  in  a  space  of  four  dimensions,  has  been  the  subject 
of  a  profound  memoir  by  Corrado  Segre  (1887)  of  Turin.  The  depth 
and  fecundity  of  this  paper  is  evinced  by  the  fact  that  a  large  pro- 
portion of  the  propositions  upon  the  plane  quartic  and  its  bitangents, 
Pascal's  theorem,  the  cubic  surface  and  its  27  lines,  Kummer's  sur- 
face and  its  configuration  of  sixteen  singular  points  and  planes,  and  on 
the  connection  between  these  figures,  are  derivable  from  propositions 
relating  to  Segre's  cubic  variety,  and  the  figure  of  six  points  or  spaces 
from  which  it  springs.  Other  investigators  into  the  properties  of  this 
beautiful  and  important  locus  in  space  of  four  dimensions  and  some  of 
its  consequences  are  G.  Castelnuovo  and  H.  W.  Richmond." 

In  1869  C.  Jordan  first  proved  "that  the  group  of  the  problem  of 
the  trisection  of  hyperelliptic  functions  of  the  first  order  is  isomorphic 
with  the  group  of  the  equation  of  the  27th  degree,  on  which  the  27 
lines  of  the  general  surface  of  the  third  degree  depend."  In  1S87  F. 
Klein  sketched  the  effective  reduction  of  the  one  problem  to  the  other, 
while  H.  Maschke,  H.  Burkhardt,  and  A.  Witting  completed  the  work 
outlined  by  Klein.  The  Galois  group  of  the  equation  of  the  27  lines 
was  investigated  also  by  L.  E.  Dickson,  F.  Kiilmen,  H.  Weber,  E. 
Pascal,  E.  Kasner  and  E.  H.  Moore. 

Surfaces  of  the  fourth  order  have  been  studied  less  thoroughly  than 
those  of  the  third.    J.  Steiner  worked  out  properties  of  a  surface  of  the 


ANALYTIC  GEOMETRY  319 

k)urtli  order  in  1S44  when  he  was  on  a  journey  to  Italy;  that  surface 
bears  this  name,  and  later  received  the  attention  of  E.  E.  Kummer. 
In  1850  Thomas  Weddle  ^  remarked  that  the  locus  of  the  vertex  of  a 
quadric  cone  passing  through  six  given  points  is  a  quartic  surface 
and  not  a  twisted  cubic  as  M.  Chasles  had  once  stated.  A.  Cayley 
gave  a  symmetric  equation  of  the  surface  in  1861.  Thereupon  Chasles 
in  1 86 1  showed  that  the  locus  of  the  vertex  of  a  cone  which  divides 
six  given  segments  harmonically  is  also  a  quartic  surface;  this  more 
general  surface  was  identified  by  Cayley  with  the  Jacobian  of  four 
quadrics,  the  Weddle  surface  corresponding  to  the  case  in  which  the 
four  quadrics  have  six  common  points.  Properties  of  the  Weddle  sur- 
face were  studied  also  by  H.  Bateman  (1905).  The  plane  section  of  a 
Weddle  surface  is  not  an  arbitrary  quartic  curve,  but  one  for  which  an 
invariant  vanishes.  Frank  Morley  proved  that  the  curve  contains  an 
infinity  of  configurations  B^,  where  it  is  cut  by  the  lines  on  the  sur- 
face. 

In  1863  and  1864,  E.  E.  Kummer  entered  upon  an  intensive  study 
of  surfaces  of  the  fourth  order.  Noted  is  the  surface  named  after  him 
which  has  16  nodes.  The  various  shapes  it  can  assume  have  been 
studied  by  Karl  Rohn  of  Leipzig.  It  has  received  the  attention  of 
many  mathematicians,  including  A.  Cayley,  J.  G.  Darboux,  F.  Klein, 
H.  W.  Richmond,  O.  Bolza,  H.  F.  Baker,  and  J.  I.  Hutchinson.'^  It  has 
been  known  for  some  time  that  Fresnel's  wave  surface  is  a  case  of 
Rummer's  sixteen  nodal  quartic  surface;  also  it  is  known  that  the 
surface  of  a  dynamical  medium  possessing  certain  general  properties 
is  a  tyT^e  of  Rummer's  surface  which  can  he  derived  from  Fresnel's 
surface  by  means  of  a  homogeneous  strain.  Rummer's  quartic  surface 
as  a  wave  surface  is  treated  by  H.  Bateman  (1909).  The  general 
Rummer's  surface  appears  to  be  the  wave  surface  for  a  medium  of  a 
purely  ideal  character. 

F.  R.  Sharpe  and  C.  F.  Craig  of  Cornell  University  have  studied 
birational  transformations  which  leave  the  Rummer  and  Weddle 
surfaces  invariant,  by  the  application  of  a  theory  due  to  F,  Severi 
(1906). 

Quintic  surfaces  have  been  investigated  at  intervals,  since  1862, 
principally  by  L.  Cremona,  H.  A.  Schwarz,  A.  Clebsch,  M.  Noether, 
R.  Sturm,  J.  G.  Darboux,  E.  CaporaH,  A.  Del  Re,  E.  Pascal,  John  E. 
Hill  and  A.  B.  Basset.  No  serious  attempt  has  been  made  to  enumer- 
ate the  different  forms  of  these  surfaces. 

Ruled  surfaces  with  isotropic  generators  have  been  considered  by 
G.  Monge,  J.  A.  Serret,  S.  Lie  and  others.  L.  P.  Eisenhart  of  Princeton 
determines  such  a  surface  by  the  curve  in  which  it  is  cut  by  a  plane 
and  the  directions  of  the  projections  on  the  plane  of  the  generators 

'  Camh.  &•  Dublin  Math.  Jour.,  Vol.  5,  1850,  p.  6q. 

2  Consult  R.  W.  H.  T.  Hudson  (1876-1904),  Kummcrs  Quartic  Surface,  Cam- 
brirlge,  IQ05. 


320  A  HISTORY  OF  ^fATHEMATICS 

of  the  surface.  In  this  way  a  ruled  surface  of  this  t_\pe  is  determined 
by  a  set  of  lineal  elements,  in  a  plane,  depending  on  one  parameter. 

While  the  classification  of  cubic  curves  was  given  by  I.  Newton, 
and  their  general  theory  was  well  under  way  two  centuries  ago,  the 
theory  of  quartir  curves  was  not  pursued  vigorously  until  Ihe  time 
of  J.  Steiner  and  L.  O.  Hesse.  Neglecting  the  classification  of  quartic 
curves  due  to  L.  Euler,  G.  Cramer  and  E.  Waring,  new  classifications 
have  been  made,  either  according  to  their  genus  (Geschlecht)  3,  2,  1,0, 
or  according  to  topologic  considerations  studied  by  A.  Cayley  in 
1865,  H.  G.  Zeuthen  (1S73),  Christian  Crone  (1877)  and  others. 
J.  Steiner  in  1855  and  L.  O.  Hesse  began  researches  on  the  28  double 
tangents  of  a  general  quartic;  24  inflections  were  found,  of  which 
G.  Salmon  conjectured  and  H.  G.  Zeuthen  proved  that  at  most  8  are 
real.  An  enumeration  (containing  nearly  200  graphs)  of  the  funda- 
mental form.s  of  quartic  curves  "when  projected  so  as  to  cut  the  line 
infinity  the  least  possible  number  of  times"  was  given  in  1896  by  Ruth 
Gentry  (1862-1917),  then  of  Bryn  Mawr  College. 

Curves  of  the  fourth  order  have  received  attention  for  many  years. 
More  recently  a  good  deal  has  been  v/ritten  on  special  curves  of  the 
fifth  order  by  Frank  Morley,  Alfred  B.  Basset,  Virgil  Snyder,  Peter 
Field,  and  others. 

Gino  Loria  of  the  University  of  Genoa,  who  has  written  extensively 
on  the  history  of  geometry,  and  the  history  of  curves  in  particular, 
has  advanced  a  theory  of  panalgebraic  curves,  which  are  in  general 
transcendental  curves.  By  definition,  a  panalgebraic  curve  must 
satisfy  a  certain  differential  equation.  A  book  of  reference  on  curves 
was  published  by  Gomes  Teixeira  in  1905  at  Madrid  under  the  title 
Tratado  de  las  curvas  especiales  notables. 

The  infinitesimal  calculus  was  first  appHed  to  the  determination 
of  the  measure  of  curvature  of  surfaces  by  J.  Lagrange,  L.  Euler,  and 
Jean  Baptiste  Marie  Meusnier  (1754-1793)  of  Paris,  noted  for  his 
mihtary  as  well  as  scientific  career.  Meusnier's  theorem,  relating  to 
curves  drawn  on  an  arbitrary  surface,  was  extended  by  S.  Lie  and  in 
1908  by  E.  Kasner.  The  researches  of  G.  Monge  and  E.  P.  C.  Dupin 
were  eclipsed  by  the  work  of  K.  F.  Gauss,  who  disposed  of  this  difficult 
subject  in  a  way  that  opened  new  vistas  to  geometricians.  His  treat- 
ment is  embodied  in  the  Disquisitiones  generales  circa  superficies  curvas 
(1827)  and  Untersuchungen  vber  Gcgenstdnde  der  hoheren  Geodasie  of 
1843  and  1846.  In  1827  he  established  the  idea  of  curvature  as  it  is 
understood  to-day.  Both  before  and  after  the  time  of  Gauss  various 
definitions  of  curvature  of  a  surface  had  been  advanced  by  L.  Euler, 
Meusnier,  Monge,  and  Dupin,  but  these  did  not  meet  with  general 
adoption.  From  Gauss'  measure  of  curvature  flows  the  theorem  of 
Johann  August  Grunert  (1797-1872),  professor  in  Greifswald,  and 
founder  in  1841  of  the  Archiv  der  Mathematik  und  Physik,  that  the 
aiilhmetical  mean  of  the  radii  of  curvature  of  all  normal  sections 


ANALYTIC  GEOMETRY  321 

through  a  point  is  the  radius  of  a  sphere  which  has  the  same  measure 
of  curvature  as  has  the  surface  at  that  point.  Gauss's  deduction  of 
the  formula  of  curvature  was  simpUhed  through  the  use  of  deter- 
minants by  Ilcinrich  Richard  Baltzer  (1818-1SS7)  of  Giessen.^  Gauss 
obtained  an  interesting  theorem  that  if  one  surface  be  developed 
(abgewickell)  upon  another,  the  measure  of  curv^ature  remains  un- 
altered at  each  point.  The  question  whether  two  surfaces  having 
the  same  curvature  in  corresponding  points  can  be  unwound,  one 
upon  the  other,  was  answered  by  F.  Minding  in  the  affirmative  only 
when  the  curvature  is  constant.  Surfaces  of  constant  and  negative 
curvature  were  called  p:eudo-spherical  surfaces  by  E.  Beltrami  in 
1S6S,  in  order,  as  he  says,  "to  avoid  circumlocution."  The  case  of 
variable  curvature  is  difficult,  and  was  studied  by  F.  Minding,  Joseph 
Liouville  (1S00-1882)  of  the  Polytechnic  School  in  Paris,  Ossian 
Bonnet  (1819-1892)  of  Paris.  Gauss's  measure  of  curvature,  expressed 
as  a  function  of  curvilinear  co-ordinates,  gave  an  impetus  to  the  study 
of  difi[erential-in\'ariants,  or  differential-parameters,  which  have  been 
investigated  by  C.  G.  J.  Jacobi,  C.  Neumann,  Sir  James  Cockle 
(1819-1895)  of  London,  G.  H.  Halphen,  and  elaborated  into  a  general 
theory  by  E.  Beltrami,  S.  Lie,  and  others.  Beltrami  showed  also  the 
connection  between  the  measure  of  curvature  and  the  geometric  axioms. 
In  iSoQ  Claude  Guichard  of  Rennes  announced  two  theorems  relating 
to  a  quadric  of  revolution  which  marked  a  new  epoch  in  the  theory 
of  deformation  of  surfaces.  Researches  along  this  line  by  Guichard 
and  Luigi  Bianchi  of  Pisa  are  embodied  in  the  second  edition  of 
Bianchi's  Lezioni  di  geometria  dijferenziale,  Pisa,  1902.  Another 
treatise  on  metric  differential  geometry  was  brought  out  in  1908  by 
Rcinhold  v.  Lilienthal  of  the  University  of  JMunster.  Not  only  does 
he  give  geometric  interpretations  of  the  first  and  second  derivatives 
by  means  of  the  tangent  and  the  circle  of  curv'ature,  but  he  revives 
a  notion  due  to  Abel  Transon  (1S05-1876)  of  Paris  which  gives  a 
geometric  interpretation  of  the  third  derivative  in  terms  of  the  ab- 
berancy  of  a  curve  and  the  axis  of  abberancy.  A  still  later  work  is  the 
Treatise  on  Dijferenlial  Geometry  of  Curoes  and  Surfaces  (1909)  by 
L.  P.  Eisenhart  of  Princeton  which  possesses  the  interesting  feature 
of  movable  axes  (the  so-called  "moving  trihedrals"  used  extensively 
by  J.  G.  Darboux),  applied  to  twisted  curves  as  well  as  surfaces;  he 
gives  the  four  transformations  of  surfaces  of  constant  curvature, 
due  to  N.  Hatzidakis  of  Athens,  L.  Bianchi  of  Pisa,  A.  V.  Biicklund 
of  Lund,  and  S.  Lie.  Eisenhart  developed  a  theory  of  transformations 
of  a  conjugate  system  of  curves  on  any  surface  into  conjugate  systems 
on  other  surfaces,  and  also  of  transformations  of  conjugate  nets  on 
two-dimensional  spreads  in  space  of  any  order." 

'  .\ugiist  Haas,  Vcrsuch  ciiicr  DarsteTlung  aer  Geschichie  des  Kriimmungsmasses. 
r-i'jingen,  1881. 
''■  BiUL  Am.  Math.  Sac,  Vol.  24,  19 17,  p.  68. 


32  2  A  HISTORY  OF  MATHEMATICS 

The  metric  part  of  differential  geometry  occupied  the  attention 
of  mathematicians  since  the  time  of  G.  Monge  and  K.  F.  Gauss,  and 
has  reached  a  high  degree  of  perfection.  Less  attention  has  been 
given  until  recently  to  projective  differential  geometry,  particularly 
to  the  differential  geometry  of  surfaces.  G.  H.  Halphen  started  with 
the  equation  y=/(x)  of  a  curve  and  determined  functions  of  y,  dy;dx, 
d'yidx",  etc.,  which  are  left  invariant  when  x  and  y  are  subjected  to 
a  general  projective  transformation.  His  early  formulation  of  the 
problem  is  unsymmetrical  and  unhomogeneous.  Using  a  certain 
system  of  partial  differential  equations  and  the  geometrical  theory 
of  semi-covariants,  E.  J.  Wilczynski  obtained  homogeneous  forms, 
such  forms  being  deduced  later  also  by  Halphen.^  Wilczynski  treats 
of  the  projective  differential  geometry  of  curves  and  ruled  surfaces, 
these  surfaces  being  prerequisite  for  his  theory  of  space  curves.  Wil- 
czynski treated  ruled  surfaces  by  a  system  of  two  linear  homogeneous 
differential  equations  of  the  second  order.  The  method  was  extended 
to  five-dimensional  space  by  E.  B.  Stouffer  of  the  University  of  Kan- 
sas.^ Developable  surfaces  were  studied  by  W.  W.  Denton  of  the 
University  of  Illinois.  Belonging  to  projective  differential  geometry 
are  J.  G.  Darboux's  conjugate  triple  systems  which  are  generalized 
notions  of  the  orthogonal  triple  systems.  The  projective  differential 
geometry  of  triple  systems  of  surfaces,  of  one-parameter  families  of 
space  curves  and  conjugate  nets  on  a  curved  surface,  and  allied  topics, 
were  studied  by  Gabriel  Marcus  Green  (1891-1919),  of  Harvard 
University. 

Differential  projective  geometry  of  hyperspace  was  greatly  advanced 
by  C.  Guichard  who  introduced  two  elements  depending  on  two  va- 
riables; they  are  the  reseau  and  the  congruence.  Differential  geometry 
of  hyperspace  was  greatly  enriched  since  1906  by  Corrado  Segre  of 
Turin,  and  by  other  geometers  of  Italy,  particularly  Gino  Fano  of 
Turin  and  Federigo  Enriques  of  Bologna;  ^  also  by  A.  Ranum  of 
Cornell,  C.  H.  Sisam  then  of  Illinois  and  C.  L.  E.  Moore  of  the  Massa- 
chusetts Institute  of  Technology. 

The  use  of  vector  analysis  in  differential  geometry  goes  back  to 
H.  G.  Grassmann  and  W.  R.  Hamilton,  to  their  successors  P.  G.  Tait, 
C.  Maxwell,  C.  Burali-Forti,  R.  Rothe  and  others.  These  men  have 
introduced  the  terms  "grad,"  "div,"  "rot."  A  geometric  study  of 
trajectories  with  the  aid  of  analytic  and  chiefly  contact  transforma- 
tions was  made  by  Edward  Kasner  of  Columbia  University  in  his 
Princeton  Colloquium  lectures  of  1909  on  the  "differential-geometric 
aspects  of  dynamics." 

1  See  E.  J.  Wilczynski  in  New  Haven  Colloquium,  igo6.  New  Haven,  19 10,  p.  156; 
also  his  Projective  Dijfcrcnlial  Geometry  of  Curves  and  Ruled  Surfaces,  Leipzig,  1906. 

2  Bull.  Am.  Math.  Soc,  Vol.  18,  p.  444. 

'  See  Enrico  Bompiani  in  Proceed.  5th  Intern.  Congress,  Cambridge,  Cambridge, 
1913,  Vol.  II,  p.  22. 


ANALYTIC  GEOMETRY  323 

Analysis  Siim 

Various  researches  have  been  brought  under  the  head  of  "analysis 
situs."  The  subject  was  first  investigated  by  G.  W.  Leibniz,  and 
was  later  treated  by  L.  Euler  who  was  interested  in  the  problem  to 

cross   all  of   the   seven   bridges  over  the  ^^ 

Pregel  river  at  Konigsberg  without  passing    __^:::iCll/~~3^ 

twice  over  any  one,  then  by  K.  F.  Gauss, ^C        ,-— CT 

whose  theory  of   knots    {Verschlingungen)  \jZII^' — ^-]Il7d;;^ 

has  been  employed  more  recently  by  Jo- 

hann  Benedict  Listing  (1S08-18S2)  of  Gottingen,  Oskar  Simony 
of  Vienna,  F.  Dingeldey  of  Darmstadt,  and  others  in  their  "topologic 
studies."  P.  G.  Tait  was  led  to  the  study  of  knots  by  Sir  William 
Thomson's  theory  of  vortex  atoms.  Through  Rev.  T.  P.  Kirkman 
who  had  studied  the  properties  of  polyhedra,  Tait  was  led  to  study 
knots  also  by  the  polyhedral  method;  he  gave  the  number  of  forms 
of  knots  of  the  first  ten  orders  of  knottiness.  Higher  orders  were 
treated  by  Kirkman  and  C.  N.  Little.  Thomas  Penyngton  Kirkman  ^ 
(1806-1S95)  was  born  at  Bolton,  near  Manchester.  During  boyhood 
he  was  forced  to  follow  his  father's  business  as  dealer  in  cotton  and 
cotton  waste.  Later  he  tore  away,  entered  the  University  of  Dublin, 
then  became  ^'icar  of  a  parish  in  Lancashire.  As  a  mathematician 
he  was  almost  entirely  self-taught.  He  wrote  on  pluquaternions  in- 
volving more  imaginaries  than  i,  j,  k,  on  group  theory,  on  mathe- 
matical mnemonics  producing  what  De  IVIorgan  called  "the  most 
curious  crocket  I  ever  saw,"  on  the  problem  of  the  "fifteen  school 
girls"  who  walk  out  three  abreast  for  seven  days,  where  it  is  required 
to  arrange  them  daily  so  that  no  two  shall  walk  abreast  more  than 
once.  This  problem  was  studied  also  by  A.  Cayley  and  Sylvester, 
and  is  related  to  researches  of  J.  Steiner. 

Another  unique  problem  was  the  one  on  the  coloring  of  maps,  first 
mentioned  by  A.  F.  Mobius  in  1840  and  first  seriously  considered  by 
Francis  Guthrie  and  A.  De  Morgan.  How  many  colors  are  necessary 
to  draw  any  map  so  that  no  two  countries  having  a  line  of  boundary 
in  common  shall  appear  in  the  same  color?  Four  different  colors 
are  found  experimentally  to  be  necessary  and  suflScient,  but  the  proof 
is  diflacult.  A.  Cayley  in  1878  declared  that  he  had  not  succeeded 
in  obtaining  a  general  proof.  Nor  have  the  later  demonstrations  by 
A.  B.  Kempe,  P.  G.  Tait,  P.  J.  Heawood  of  the  University  of  Durham, 
W.  E.  Story  of  Clark  University,  and  J.  Peterson  of  Copenhagen 
removed  the  difficulty.^  Tait's  proof  leads  to  the  interesting  con- 
clusion that  four  colors  may  not  be  sufficient  for  a  map  drawn  on  a 
multiply-connected  surface  like  that  of  an  anchor  ring.  Further 
studies  of  maps  on  such  surfaces,  and  of  the  problem  in  general,  are 

1  A.  Macfarlane,  Ten  British  Mathematicians,  1916,  p.  122. 

2  W.  Ahrens,  U nlerhaltungcn  iind  Spiele,  1901,  p.  340. 


324  A  HISTORY  OF  MATHEMATICS 

due  to  0.  Veblen  (191 2)  and  G.  D.  Birkhoff  (1913).  On  a  surface  of 
genus  zero  "it  is  not  known  whether  or  not  only  four  colors  always 
suffice."  A  similar  question  considers  the  maximum  possible  number 
of  countries,  when  every  country  touches  every  other  along  a  line. 
Lothar  Hefftcr  wrote  on  this  conundrum,  in  1S91  and  again  in  later 
articles,  as  did  also  A.  B.  Kempe  and  others.  In  the  hands  of  Riemann 
the  analysis  situs  had  for  its  object  the  determination  of  what  remains 
unchanged  under  transformations  brought  about  by  a  combination 
of  infinitesimal  distortions.  In  continuation  of  his  work,  Walter  Dyck 
of  Munich  wrote  on  the  analysis  situs  of  tliree-dimensional  spaces. 
Researches  of  this  sort  have  important  bearings  in  modern  mathe- 
matics, particularly  in  connection  with  correspondences  and  differ- 
ential equations.-^ 

Intrinsic  Co-ordinates 

As  a  reaction  against  the  use  of  the  arbitrary  Cartesian  and  polar 
co-ordinates  there  came  the  suggestion  from  the  philosophers  K.  C.  F. 
Krause  (1781-1832),  A.  Peters  (1S03-1S76)  that  magnitudes  inherent 
to  a  curve  be  used,  such  as  s,  the  length  of  arc  measured  from  a  fixed 
point,  and  (f,  the  angle  which  the  tangent  at  the  end  of  5  makes  with 
a  fixed  tangent.  William  Whewell  (1794-1S66)  of  Cambridge,  the 
2iu\hor  oi  the  History  of  the  Inductive  Sciences,  1S37-1838,  introduced  in 
1S49  the  name  "intrinsic  equation"  and  pointed  out  its  use  in  study- 
ing successive  evolutes  and  involutes.  The  method  was  used  by  Wil- 
liam Walton  (1S13-1901)  of  Cambridge,  J.  J.  Sylvester  in  1868,  J. 
Casey  in  1866,  and  others.  Instead  of  using  5  and  <p,  other  writers 
have  introduced  the  radius  of  curvature  p,  and  have  used  either  ^ 
and  p,  or  (f  and  p.  The  co-ordinates  {cp,  p)  were  employed  by 
L.  Euler  and  several  nineteenth  century  mathematicians,  but  alto- 
gether the  co-ordinates  (s,  p)  have  been  used  most.  The  latter  were 
used  by  L.  Euler  in  1741,  by  Sylvestre  Frangois  Lacroix  (1765-1843), 
by  Thomas  Hill  (1818-1891,  who  was  at  one  time  president  of  Harvard 
College),  and  in  recent  years  especially  by  Ernesto  Cesaro  of  the 
University  of  Naples  who  published  in  1S96  his  Geometria  intrinseca 
which  was  translated  into  German  in  1901  by  G.  Kowalewski  uiider 
the  title,  Vorlesungen  ilber  natilrliche  Gcometrie}  Researches  along 
this  Une  are  due  also  to  Amedee  Mannheim  (1S31-1906)  of  Paris,  the 
designer  of  a  well-known  slide  rule. 

The  application  of  intrinsic  or  natural  co-ordinates  to  surfaces  is 
less  common.  Edward  Kasner  ^  said  in  1904  that  in  the  "theory  of 
surfaces,  natural  co-ordinates  may  be  introduced  so  as  to  fit  into  the 

1  See  J.  Hadamard,  Four  Lectures  on  Mathematics  delivered  at  Colmnhia  Uuivrrsity 
in  igii,  New  York,  1915,  Lecture  III. 

*Our  information  is  drawn  from  E.  WolfEng's  article  on  "Naturliche  Koor- 
dinaten"  in  Bihliolhcca  mathcviatica,  3.  S.,  Vol.  I,  1900,  pp.  142-159. 

'  Bull.  Am.  Math.  Soc,  Vol.  11,  1905,  p.  303. 


ANALYTIC  GEOMETRY  325 

so-called  geometry  of  a  flexible  but  inextcnsible  surface,  originated  by 
K.  F.  Gauss,  in  which  the  criterion  of  equivalence  is  applicability  or, 
according  to  the  more  accurate  jjhraseology  of  A.  Voss,  isometry. 
Intrinsic  co-ordinates  must  then  be  invariant  with  respect  to  bend- 
ing. .  .  .  The  simplest  exami)le  of  a  complete  isometric  group  is 
the  group  ty])ificd  by  the  plane,  consisting  of  all  the  developable  sur- 
faces. In  this  case  the  equations  of  the  group  may  be  obtamed  ex- 
])iicit]y,  in  terms  of  eliminations,  differentiations  and  quadratures.  .  .  . 
Until  the  year  1S66,  not  a  single  case  analogous  to  that  of  the  de- 
velopable surfaces  was  discovered.  Julius  Weingarten,  by  means  of 
his  theory  of  evolutcs,  then  succeeded  in  determining  the  complete 
group  of  the  catenoid  and  of  the  paraboloid  of  revolution,  and,  some 
twenty  years  later,  a  fourth  group  defined  in  terms  of  minimal  sur- 
faces. During  the  past  decade,  the  French  geometers  have  concen- 
trated their  efforts  in  this  field  mainly  on  the  arbitrary  paraboloid 
(and  to  some  extent  on  the  arbitrary  quadric).  The  difficulties  even 
in  this  extremely  restricted  and  apparently  simple  case  are  great, 
and  are  only  gradually  being  conquered  by  the  use  of  almost  the  whole 
wealth  of  niodcrn  analysis  and  the  invention  of  new  methods  which 
undoubtedly  have  wider  fields  of  application.  The  results  obtained 
exhibit,  for  example,  connections  with  the  theories  of  surfaces  of 
constant  curvature,  isometric  surfaces,  Backlund  transformations, 
and  motions  with  two  degrees  of  freedom.  The  principal  workers 
are  J.  G.  Darboux,  E.  J.  B.  Goursat,  L.  Bianchi,  A.  L.  Thybaut,  E. 
Cosserat,  IM.  G.  Servant,  C.  Guichard,  and  L.  RafTy." 

Definition  of  a  Curve 

The  theory  of  sets  of  points,  originated  by  G.  Cantor,  has  given 
rise  to  new  views  on  the  theory  of  curves  and  on  the  meaning  of  con- 
tent. What  is  a  curve?  Camille  Jordan  in  his  Coiirs  d' analyse  defined 
it  tentatively  as  a  "continuous  line."  W.  H.  Young  and  Grace 
Chisholm  Young  in  their  Theory  of  Sets  of  Points,  1906,  p.  222,  define 
a  "Jordan  curve"  as  "a  plane  set  of  ])oints  which  can  be  brought  into 
continuous  (i,  i)  correspondence  with  the  points  of  a  closed  segment 
(a,  z)  of  a  straight  line."  A  circle  is  a  closed  Jordan  curve.  Jordan 
asked  the  question,  whether  it  was  possible  for  a  curve  to  fill  up  a 
space.  G.  Peano  answered  that  a  "continuous  line"  may  do  so  and 
constructed  in  Math.  Annalen,  Vol.  36,  1890,  the  so-called  "space- 
filling curve"  (the  "Peano  curve")  to  fortify  his  assertion.  His  mode 
of  construction  has  been  modified  in  several  ways  since.  The  most 
noted  of  these  are  due  to  E.  H.  Moore  ^  and  D.  Hilbert.  In  1916  R.  L. 
Aloore  of  the  University  of  Pennsylvania  proved  that  every  two  points 
of  a  continuous  curve,  no  matter  how  crinkly,  can  be  joined  by  a 
simple  continuous  arc  that  lies  wholly  in  the  curve.  As  it  dees  not 
'  Trans.  Am.  Math.  Soc,  Vol.  I,  190c,  pp.  72-90. 


326  A  HISTORY  OF  MATHEMATICS 

seem  desirable  to  depart  from  our  empirical  notions  so  far  as  to  alicw 
the  term  curve  to  be  applied  to  a  region,  more  restricted  definitions 
of  it  become  necessary.  C.  Jordan  demanded  that  a  curve  x=(p{t), 
y=\p{t)  should  have  no  double  points  in  the  interval  a^t<b.  Schon- 
flies  regards  a  curve  as  the  frontier  of  a  region.  0.  Veblen  defines  it 
in  terms  of  order  and  linear  continuity.  W.  H.  Young  and  Grace  C. 
Young  in  their  Theory  of  Sets  of  Points  define  a  curve  as  a  plane  set 
of  points,  dense  nowhere  in  the  plane  and  bearing  other  restrictions, 
yet  such  that  it  may  consist  of  a  net-work  of  arcs  of  Jordan  curves. 

Other  curves  of  previously  unheard  of  properties  were  created  as 
the  result  of  the  generaUzation  of  the  function  concept.     The  con- 

n=00 

tinuous  curve  represented  by  y=  2  i>"-  cos  Tr(a"'x),  where  a  is  an  even 

n  =o 

integer  >i,  6  a  real  positive  number  <i,  was  shown  by  Weierstrass  ^ 
to  possess  no  tangents  at  any  of  its  points  when  the  product  ab  exceeds 


a  certain  limit;  that  is,  we  have  here  the  startling  phenomenon  of  a 
continuous  function  which  has  no  derivative.  As  Christian  Wiener 
explained  in  iS8i,  this  curve  has  countless  oscillations  within  every 
finite  interval.  An  intuitively  simpler  curve  was  invented  by  Helge 
v.  Koch  of  the  University  of  Stockholm  in  1904  {Acta  math.,  Vol.  30, 
1906,  p.  145)  which  is  constructed  by  elementary  geometry,  is  con- 
tinuous, yet  has  no  tangent  at  any  of  its  points;  the  arc  between  any 
two  of  its  points  is  infinite  in  length.  While  this  curve  has  been  repre- 
sented analytically,  no  such  representation  has  yet  been  found  for 
the  so-called  H-curve  of  Ludwig  Boltzmann  (1844-1906)  of  Vienna, 
in  Math.  Annalen,  Vol.  50,  1S98,  which  is  continuous,  yet  tangentless. 
The  adjoining  figure  shows  its  construction.  Boltzmann  used  it  to 
visualize  theorems  in  the  theory  of  gases. 

Fundamental  Postulates 

The  foundations  of  mathematics,  and  of  geometry  in  particular, 
received  marked  attention  in  Italy.    In  18S9  G.  Peano  took  the  no\-el 

1  P.  du  Bois-Reymond  "  Versuch  einer  Klassification  der  willkiirllchen  I'unk- 
tionen  reeller  Argumente,"  Crelle,  Vol.  74,  1874,  p.  ag. 


ANALYTIC  GEOMETRY  327 

view  that  geometric  elements  are  mere  things,  and  laid  down  the 
principle  that  there  should  be  as  few  undefined  symbols  as  possible. 
In  1897-9  his  pupil  Mario  Fieri  (1860-1904)  of  Catania  used  only  two 
undefined  symbols  for  projective  geometry  and  but  two  for  metric 
geometry.  In  1S94  Pcano  considered  the  independence  of  axioms. 
By  1S97  the  Italian  mathematicians  had  gone  so  far  as  to  make  it  a 
postulate  that  points  are  classes.  These  fundamental  features  elab- 
orated by  the  Italian  school  were  embodied  by  David  Hilbert  of 
Gottingen,  along  with  important  novel  considerations  of  his  owti,  in 
his  famous  Grundlagen  der  Geomctrie,  1899.  A  fourth  enlarged  edition 
of  this  appeared  in  19 13.  E.  B.  Wilson  says  in  praise  of  Hilbert:  "The 
archimedean  axiom,  the  theorems  of  B.  Pascal  and  G.  Desargues,  the 
anal)'sis  of  segments  and  areas,  and  a  host  of  things  are  treated  either 
for  the  first  time  or  in  a  new  way,  and  with  consummate  skill.  We 
should  say  that  it  was  in  the  technique  rather  than  in  the  philosophy 
of  geometry  that  Hilbert  created  an  epoch."  ^  In  Hilbert's  space  of 
1899  are  not  all  the  points  which  are  in  our  space,  but  only  those  that, 
starting  from  two  given  points,  can  be  constructed  with  ruler  and 
compasses.  In  his  space,  remarks  Poincare,  there  is  no  angle  of  10°. 
So  in  the  second  edition  of  the  Grundlagen,  Hilbert  introduced  the 
assumption  of  completeness,  which  renders  his  space  and  ours  the 
same.  Interesting  is  Hilbert's  treatment  of  non-Archimedean  geom- 
etry where  all  his  assumptions  remain  true  save  that  of  Archimedes, 
and  for  which  he  created  a  system  of  non-Archimedean  numbers. 
This  non-Archimedean  geometry  was  first  conceived  by  Giuseppe 
Veronese  (1854-1917),  professor  of  geometry  at  the  University  of 
Padua.  Our  common  space  is  only  a  part  of  non-Archimedean 
space.  Non-Archimedean  theories  of  proportion  were  given  in  1902  by 
A.  Kneser  of  Breslau  and  in  1904  by  P.  J.  Mollerup  of  Copenhagen. 
Hilbert  devoted  in  his  Grundlagen  a  chapter  to  Desargues'  theorem. 
In  1902  F.  R.  Moulton  of  Chicago  outlined  a  simple  non-Desarguesian 
plane  geometry. 

In  the  United  States,  George  Bruce  Halsted  based  his  Rational 
Geometry,  1904,  upon  Hilbert's  foundations.  A  second,  revised  edition 
appeared  in  1907.  One  of  Hilbert's  pupils.  Max  W.  Dehn,  showed 
that  the  omission  of  the  axiom  of  Archimedes  (Eudoxus)  gives  rise 
to  a  semi-Euclidean  geometry  in  which  similar  triangles  exist  and  their 
sum  is  two  right  angles,  yet  an  infinity  of  parallels  to  any  straight 
line  may  be  drawn  through  any  given  point. 

Systems  of  axioms  upon  which  to  build  projective  geometry  were 
first  studied  more  particularly  by  the  Italian  school — G.  Peano,  M. 
Pieri,  Gino  Fano  of  Turin.  This  subject  received  the  attention  also 
of  Theodor  Vahlen  of  Vienna  and  Friedrich  Schur  of  Strassburg. 
.\xioms  of  descriptive  geometry  have  been  considered  mainly  by 

1  Bull.  Am.  Math.  Soc,  Vol.  11,  1904,  p.  77.  Our  remarks  on  the  Italian  school 
are  drawn  from  Wilson's  article. 


328  A  HISTORY  OF  MATHEMATICS 

Italian  and  American  mathematicians,  and  by  D.  Hilbert.  The 
introduction  of  order  was  achieved  by  G.  Peano  by  taking  the  class 
of  points  which  he  between  any  two  points  as  the  fundamental  idea, 
by  G.  Vailati  and  later  by  B.  Russell,  on  the  fundamental  conception 
of  a  class  of  relations  or  class  of  points  on  a  straight  line,  by  O.  Veblen 
(1904)  on  the  study  of  the  properties  of  one  single  three-term  relation 
of  order.  A.  N.  Whitehead^  refers  to  O.  Veblen's  method:  "This 
method  of  conceiving  the  subject  results  in  a  notable  simplification, 
and  combines  advantages  from  the  two  previous  methods."  While 
D.  Hilbert  has  six  undefined  terms  (point,  straight  line,  plane,  between, 
parallel,  congruent)  and  twenty-one  assumptions,  Veblen  gives  only 
two  undefined  terms  (point,  between)  and  only  twelve  assumptions. 
However,  the  derivation  of  fundamental  theorems  is  somewhat 
harder  by  Veblen's  axioms.  R.  L.  Moore  showed  that  any  plane 
satisfying  Veblen's  axioms  I-VIII,  XI  is  a  number-plane  and  con- 
tains a  system  of  continuous  curves  such  that,  with  reference  to  these 
curves  regarded  as  straight  lines,  the  plane  is  an  ordinary  Euclidean 
plane. 

In  1907,  Oswald  Veblen  and  J.  W.  Young  gave  a  completely  in- 
dependent set  of  assumptions  for  projective  geometry,  in  which  points 
and  undefined  classes  of  points  called  lines  have  been  taken  as  the 
undefined  elements.  Eight  of  these  assumptions  characterize  general 
projective  spaces;  the  addition  of  a  ninth  assumption  yields  properly 
projective  spaces.^ 

Axioms  for  line  geometry  based  upon  the  "line"  as  an  undefined 
element  and  "intersection"  as  an  undefined  relation  between  un- 
ordered pairs  of  elements,  were  given  in  1901  by  M.  Fieri  of  Catania, 
and  in  simpler  form,  in  1914  by  E.  R.  Hedrick  and  L.  Ingold  of  the 
University  of  Missouri. 

Text-books  built  upon  some  such  system  of  axioms  and  possessing 
great  generality  and  scientific  interest  have  been  written  by  Federigo 
Enriques  of  the  University  of  Bologna  in  1898,  and  by  O.  Veblen 
and  J.  W.  Young  in  1910. 

Geometric  Models 

Geometrical  models  for  advanced  students  began  to  be  manu- 
factured about  1879  by  the  firm  of  L.  Brill  in  Darmstadt.  Many 
of  the  early  models,  such  as  Kummer's  surface,  twisted  cubics,  the 
tractrix  of  revolution,  were  made  under  the  direction  of  F.  Klein  and 
Alexander  von  Brill.  Since  about  1890  this  firm  developed  into  that 
of  Martin  Schilling  (1866-1908)  of  Leipzig.  The  catalogue  of  the 
firm  for  191 1  described  some  400  models.  Since  1905  the  firm  of 
B.  G.  Teubner  in  Leipzig  has  offered  models  designed  by  Herman" 

^  The  Axioms  of  Descriptive  Geometry,  Cambridge,  1907,  p.  2. 
2  Bull.  Am.  Math.  Soc,  Vol.  14,  1908,  p.  251. 


ALGEBRA  329 

Wiener,  many  of  which  are  intended  for  secondary  instruction.  \'al- 
uable  in  this  connection  is  the  Katalog  mathematischer  uiid  mathe- 
matisch-physikaUscher  Modelle,  Apparate  wid  Instrumenle  by  Walter 
V.  Dyck,  professor  in  Munich.  At  the  Napier  tercentenary  celebra- 
tion in  Edinburgh,  in  1914,  Crum  Brown  exhibited  models  of  various 
sorts,  including  models  of  cubic  and  quartic  surfaces,  interlacing  sur- 
faces, regular  solids  and  related  forms,  and  thermodynamic  models; 
D.  M.  Y.  Sommerville  displayed  models  of  the  projection,  on  three- 
dimensional  space,  of  a  four-dimensional  figure;  Lord  Kelvin's  tide- 
calculating  machine  illustrated  the  combination  of  simple  harmonic 
motions.^ 

Algebra 

The  progress  of  algebra  in  recent  times  may  be  considered  under 
three  principal  heads:  the  study  of  fundamental  laws  and  the  birth 
of  new  algebras,  the  growth  of  the  theory  of  equations,  and  the  de- 
velopment of  what  is  called  modern  higher  algebra. 

The  general  theory  of  a*,  where  both  a  and  b  are  complex  numbers, 
was  outlined  by  L.  Euler  in  1749  in  his  paper,  Recherches  sur  les  racines 
imaginaries  des  equations,  but  it  failed  to  command  attention.  At 
the  beginning  of  the  nineteenth  century  the  theory  of  the  general 
power  was  elaborated  in  Germany,  England,  France  and  the  Nether- 
lands. In  the  early  history  of  logarithms  of  positive  numbers  it  was 
found  surprising  that  logarithms  were  defined  independently  of  ex- 
ponents. Now  we  meet  a  second  surprise  in  finding  that  the  theory 
of  a*  is  made  to  depend  upon  logarithms.  Historically  the  logarithmic 
concept  is  the  more  primitive.  The  general  theory  of  a*  was  developed 
by  Martin  Ohm  (1792-1872),  professor  in  Berlin  and  a  brother  of 
the  physicist  to  whom  we  owe  "Ohm's  law."  Martin  Ohm  is  the 
author  of  a  much  criticised  series  of  books,  Versuch  eines  vollkommen 
consequenten  Systems  der  Maihcmatik,  Nurnberg,  1822-1852.  Our 
topic  was  treated  in  the  second  volume,  dated  1823,  second  edition 
1829.  After  having  developed  the  Eulerian  theory  of  logarithms  Ohm 
takes  up  a^,  where  a=p+qi,  x=  a+^i.  Assuming  e^  as  always  single- 
valued  and  letting  v=\/p^+q^,  log  c=Lv+{=t: 2mir+(p)i,  he  takes 

^i^gajloga^^aLv-^    (zi=  2mTr  +  4>).     { coS    [/3.    Lv+ a      (±2    W    Tr+(f))]+i. 

sin  [/3Lv-l-a(=fc  2mT+(f})]  |,  where  m=o,  +i,  +2,  .  .  .  and  L.  signifies 
the  tabular  logarithm.  Thus  the  general  power  has  an  infinite  number 
of  values,  but  all  are  of  the  form  a+bi.  Ohm  shows  (i)  that  all  of  the 
values  (infinite  in  number)  are  equal  when  .r  is  an  integer,  (2)  that  there 
are  n  distinct  values  when  x  is  a  real,  rational  fraction,  (3)  that  some  of 
the  values  are  equal,  though  the  number  of  distinct  values  is  infinite, 
when  X  is  real  but  irrational,  (4)  that  the  values  are  all  distinct  when  x  is 
imaginary.  He  inquires  next,  how  the  formulas  (A)  a^.av  =  a'+y,  (B)c^ 
>  Consult  E.  M.  Horsburgh  Handbook  of  the  Exhibition  of  Napier  relics,  etc.,  1914, 
p.  302. 


,^3o  A  HISTORY  OF  MATHEMATICS 

^av^a^'-y,  (C)  a^.b^={ab)^,  (D)  a'^h^  =  {a^h)  ,  (E)  {a'')y^a^^  apply 
to  the  general  exponent  c^,  and  finds  that  (A),  (FJ  and  (E)  are  incom- 
plete equations,  since  the  left  members  have  "many,  many  more" 
values  than  the  right  members,  although  the  right-hand  values  (infinite 
in  ntmiber)  are  all  found  among  tne  "infinite  times  infinite"  values  on 
the  left;  that  (C)  and  (D)  are  complete  equations  for  the  general  case. 
A  failure  to  recognize  that  equation  E  is  incomplete  led  Thomas 
Clausen  (1801-1885)  of  Altona  to  a  paradox  (Crelle's  Journal,  Vol.  2, 
1827,  p.  286)  which  was  stated  by  E.  Catalan  in  1869  in  more  con- 
densed form,  thus:  e2«7r/=g2«7r/^  where  m  and  n  are  distinct  integers. 
Raising  both  sides '  to  the  power  A ,  there  results  the  absurdity, 
e-mir=e—mr^  Ohm  introduced  a  notation  to  designate  some  particular 
value  of  a^,  but  he  did  not  introduce  especially  the  particular  value 
which  is  now  called  the  "principal  value."  Otherwise  his  treatment 
of  the  general  power  is  mainly  that  of  the  present  time,  except,  of 
course,  in  the  explanation  of  the  irrational.  From  the  general  power 
Ohm  proceeds  to  the  general  logarithm,  having  a  complex  number 
as  its  base.  It  is  seen  that  the  Eulerian  logarithms  served  as  a  step- 
ladder  leading  to  the  theory  of  the  general  power;  the  theory  of  the 
general  power,  in  turn,  led  to  a  more  general  theory  of  logarithms 
having  a  complex  base. 

The  Philosophical  Transactions  (London,  1829)  contain  two  articles 
on  general  powers  and  logarithms — one  by  John  Graves,  the  other 
by  John  Warren  of  Cambridge.  Graves,  then  a  young  man  of  23, 
was  a  class-fellow  of  William  Rowan  Hamilton  in  Dublin.  Graves 
became  a  noted  jurist.  Hamilton  states  that  reflecting  on  Graves's 
ideas  on  imaginaries  led  him  to  the  invention  of  quaternions.  Graves 
obtains  log  i  =  (2m'Tri)/(i+2mTi).  Thus  Graves  claimed  that  gen- 
eral logarithms  involve  two  arbitrary  integers,  m  and  m',  instead 
of  simply  one,  as  given  by  Euler.  Lack  of  explicitness  involved 
Graves  in  a  discussion  with  A.  De  Morgan  and  G.  Peacock,  the  out- 
come of  which  was  that  Graves  withdrew  the  statement  contained 
in  the  title  of  his  paper  and  implying  an  error  in  the  Eulerian  theory, 
while  De  Morgan  admitted  that  if  Graves  desired  to  extend  the  idea 
of  a  logarithm  so  as  to  use  the  base  ei-f-swir/^  there  was  no  error  in- 
volved in  the  process.  Similar  researches  were  carried  on  by  A.  J.  H. 
Vincent  at  Lille,  D.  F.  Gregory,  De  Morgan,  W.  R.  Hamilton  and 
G.  M.  Pagani  (1796-1855),  but  their  general  logarithmic  systems, 
involving  complex  numbers  as  bases,  failed  of  recognition  as  useful 
mathematical  inventions.^    We  pause  to  sketch  the  life  of  De  Morgan. 

Augustus  De  Morgan  (1806-1871)  was  born  at  Madura  (Madras), 
and  educated  at  Trinity  College,  Cambridge.  For  the  determination 
of  the  year  of  his  birth  (assumed  to  be  in  the  nineteenth  century)  he 
proposed  the  conundrum,  "I  was  x  years  of  age  in  the  year  x'^."    His 

1  For  references  and  fuller  details  see  F.  Cajori  in  Am.  Math.  Monthly,  Vol.  20, 
1913,  PP-  175-182. 


ALGEBRA  331 

scruples  about  the  doctrines  of  the  estabUshed  church  prevented  hrm 
from  proceeding  to  the  M.  A.  degree,  and  from  sitting  for  a  fellowship. 
It  is  said  of  him,  ''he  never  voted  at  an  election,  and  he  never  visited 
the  House  of  Commons,  or  the  Tower,  or  Westminster  Abbey."  In 
1S28  he  became  professor  at  the  newly  established  University  of 
London,  and  taught  there  until  1867,  except  for  five  years,  from  183 1- 
1S35.  He  was  the  first  president  of  the  London  Mathematical  Society 
which  was  founded  in  1S66.  De  Morgan  was  a  unique,  manly  char- 
acter, and  pre-eminent  as  a  teacher.  The  value  of  his  original  work  lies 
not  so  much  in  increasing  our  stock  of  mathematical  knowledge  as  in 
putting  it  all  upon  a  more  logical  basis.  He  felt  keenly  the  lack  of  close 
reasoning  in  mathematics  as  he  received  it.  He  said  once:  "We  know 
that  mathematicians  care  no  more  for  logic  than  logicians  for  mathe- 
matics. The  two  eyes  of  exact  science  are  mathematics  and  logic: 
the  mathematical  sect  puts  out  the  logical  eye,  the  logical  sect  puts 
out  the  mathematical  eye;  each  believing  that  it  can  see  better  with 
one  eye  than  with  two."  De  Morgan  analyzed  logic  mathematically, 
and  studied  the  logical  analysis  of  the  laws,  symbols,  and  operations 
of  mathematics;  he  wrote  a  Formal  Logic  as  well  as  a  Double  Algebra, 
and  corresponded  both  with  Sir  William  Hamilton,  the  metaphysician, 
and  Sir  William  Rowan  Hamilton,  the  mathematician.  Few  con- 
temporaries were  as  profoundly  read  in  the  history  of  mathematics 
as  was  De  IVIorgan.  No  subject  was  too  insignificant  to  receive  his 
attention.  The  authorship  of  "Cocker's  Arithmetic"  and  the  work 
of  circle-squarers  was  investigated  as  minutely  as  was  the  history  of 
the  calculus.  Numerous  articles  of  his  lie  scattered  in  the  volumes  of 
the  Penny  and  English  Cyclop(Bdias.  In  the  article  "Induction  (Mathe- 
matics)," first  printed  in  1838,  occurs,  apparently  for  the  first  time, 
the  name  "mathematical  induction";  it  was  adopted  and  popularized 
by  I.  Todhunter,  in  his  Algebra.  The  term  "  induction"  had  been  used 
by  John  WaUis  in  1656,  in  his  Arithmetica  infinitorum;  he  used  the 
"induction"  known  to  natural  science.  In  1686  Jacob  Bernoulli 
criticised  him  for  using  a  process  which  was  not  binding  logically  and 
then  advanced  in  place  of  it  the  proof  from  n  to  n-\- 1 .  This  is  one  of  the 
several  origins  of  the  process  of  mathematical  induction.  From  WaUis 
to  De  Morgan,  the  term  "induction"  was  used  occasionally  in  mathe- 
matics, and  in  a  double  sense,  (i)  to  indicate  incomplete  inductions  of 
the  kind  known  in  natural  science,  (2)  for  the  proof  from  n  to  n-\-i.  De 
Morgan's  "mathematical  induction"  assigns  a  distinct  name  for  the 
latter  process.  The  Germans  employ  more  commonly  the  name  "  voll- 
standige  Induktion,"  which  became  current  among  them  after  the  use 
of  it  by  R.  Dedekind  in  his  Was  slnd  utid  was  sollen  die  Zahlen,  1887. 
De  Morgan's  Dijjerential  Calcidus,  1842,  is  still  a  standard  work,  and 
contains  much  that  is  original  with  the  author.  For  the  Encyclopoedin 
MctropoUtana  he  wrote  on  the  Calculus  of  Functions  (giving  principles 
of  symbolic  reasoning)  and  on  the  theor}'  of  probability.    In  the  Cal- 


332  A  HISTORY  OF  MATHEMATICS 

cuius  of  Functions  he  proposes  the  use  of  the  slant  line  or  "solidus" 
for  printing  fractions  in  the  text;  this  proposal  was  adopted  by  G.  G. 
Stokes  in  iSSo.^  Cayley  wrote  Stockes,  "I  think  the  'solidus'  looks 
very  well  indeed  .  .  .  ;  it  would  give  you  a  strong  claim  to  be  President 
of  a  Society  for  the  prevention  of  Cruelty  to  Printers."  ^ 

Celebrated  is  De  Morgan's  Budget  of  Paradoxes,  London,  1872,  a 
second  edition  of  which  was  edited  by  David  Eugene  Smith  in  1915. 
De  Morgan  published  memoirs  "On  the  Foundation  of  Algebra" 
in  the  Trans,  of  the  Cambridge  Phil.  Soc.,  1841,  1842,  1844  and  1847. 

The  ideas  of  George  Peacock  and  De  Morgan  recogni  e  the  possi- 
bility of  algebras  which  differ  from  ordinary  algebra.  Such  algebras 
were  indeed  not  slow  in  forthcoming,  but,  like  non-Euclidean  geometry, 
some  of  them  were  slow  in  finding  recognition.  This  is  true  of  H.  G. 
Grassmann's,  G.  Bellavitis's,  and  B.  Peirce's  discoveries,  but  W.  R. 
Hamilton's  quaternions  met  with  immediate  appreciation  in  England. 
These  algebras  offer  a  geometrical  interpretation  of  imaginaries. 

William  Rowan  Hamilton  (i 805-1 865)  was  born  of  Scotch  parents 
in  Dublin.  His  early  education,  carried  on  at  home,  was  mainly  in 
languages.  At  the  age  of  thirteen  he  is  said  to  have  been  familiar  with 
as  many  languages  as  he  had  lived  years.  About  this  time  he  came 
across  a  copy  of  I.  Newton's  Universal  Arithmetic.  After  reading  that, 
he  took  up  successively  analytical  geometry,  the  calculus,  Newton's 
Principia,  Laplace's  Mrcanique  Celeste.  At  the  age  of  eighteen  he  pub- 
lished a  paper  correcting  a  mistake  in  Laplace's  work.  In  1824  he 
entered  Trinity  College,  Dublin,  and  in  1827,  while  he  was  still  an 
undergraduate,  he  was  appointed  to  the  chair  of  astronomy.  C.  G.  J. 
Jacobi  met  Hamilton  at  the  meeting  of  the  British  Association  at 
Manchester  in  1842  and,  addressing  Section  A,  called  Hamilton  "le 
Lagrange  de  votre  pays."  Hamilton's  early  papers  were  on  optics. 
In  1832  he  predicted  conical  refraction,  a  discovery  by  aid  of  mathe- 
matics which  ranks  with  the  discovery  of  Neptune  by  U.  J.  J.  Le 
Verrier  and  J.  C.  Adams.  Then  followed  papers  on  the  Principle  of 
Varying  Action  (1827)  and  a  general  method  of  dynamics  (1834-1835). 
He  wrote  also  on  the  solution  of  equations  of  the  fifth  degree,  the 
hodograph,  fluc'.uating  functions,  the  numerical  solution  of  differential 
equations. 

The  capital  discovery  of  Hamilton  is  his  quaternions,  in  which  his 
study  of  algebra  culminated.  In  1835  he  published  in  the  Transactions 
of  I  he  Royal  Irish  A  cademy  his  Theory  of  Algebraic  Couples.  He  re- 
garded algebra  "as  being  no  mere  art,  nor  language,  nor  primarily 

'  G.  G.  Stokes,  Malh.  and  Pliys.  Papers,  Vol.  I.  Cambridge,  1880,  p.  vii;  see  also 
J.  Larmor,  Memoir  and  Scic.  Corr.  of  G.  G.  Stokes,  Vol.  I,  1907,  p.  397. 

2  An  earlier  use  of  the  solidus  in  designating  fractions  occurs  in  one  of  the  very 
fi  st  text  books  published  in  California,  viz.,  the  Definicion  de  las  princi pales 
opcraciones  dc  arismetica  by  Henri  Cambuston,  26  pages  printed  at  Monterey  ia 
184.5.     The  solidus  appears  slightly  curved 


ALGEBRA  7^3 

a  science  of  quantity,  but  rather  as  the  science  of  ordei  of  progres- 
sion." Time  appeared  to  him  as  the  picture  of  such  a  progression. 
Hence  his  definition  of  algebra  as  "the  science  of  pure  time."  It  was 
the  subject  of  years'  meditation  for  him  to  determine  what  he  should 
regard  as  the  product  of  each  pair  of  a  system  of  perpendicular  directed 
lines.  At  last,  on  the  i6th  of  October,  1S43,  while  walking  with  his 
wife  one  evening,  along  the  Royal  Canal  in  Dublin,  the  discovery  of 
cjuaternions  flashed  upon  him,  and  he  then  engraved  with  his  knife 
on  a  stone  in  Brougham  Bridge  the  fundamental  formula  i-=f=k'  = 
ijk=  —  i.  At  the  general  meeting  of  the  Irish  Academy,  a  month 
later,  he  made  the  first  communication  on  quaternions.  An  account 
of  the  discovery  was  given  the  following  year  in  the  Philosophical 
Magazine.  Hamilton  displayed  wonderful  fertility  in  their  develop- 
ment. His  Lectures  on  Quaternions,  delivered  in  Dublin,  were  printed 
in  1852. 

In  1S5S  P.  G.  Tait  was  introduced  to  Hamilton  and  a  correspond- 
ence was  carried  on  between  them  which  brought  Hamilton  back  to 
the  further  development  of  quaternions  along  the  lines  of  quaternion 
differentials,  the  linear  vector  function  and  A.  Fresnel's  wave  surface, 
and  led  him  to  prepare  the  Elements  of  Quaternions,  1866,  which  he 
did  not  live  to  complete.  Only  500  copies  were  printed.  A  new 
edition  has  been  published  recently  by  Charles  Jasper  Joly  (1864- 
1906),  his  successor  in  Dunsink  Observatory.  Tait's  own  Elementary 
Treatise  on  Quaternions  was  projected  in  1859,  but  was  withheld  from 
publication  until  Hamilton's  work  should  appear;  it  was  finally  pub- 
lished in  1S67.  P.  G.  Tait's  chief  accomplishment  was  the  develop- 
ment of  the  operator  v,  "which  was  done  in  the  later,  greatly  enlarged 
editions.^  Tait  submitted  his  quaternionic  theorems  to  the  judgment 
of  Clerk  Maxwell,  and  Maxwell  came  to  recognize  the  power  of  the 
quaternion  calculus  in  dealing  with  physical  problems.  "Tait  brought 
out  the  real  physical  significance  of  the  quantities  SyO",  Vvo",  V^- 
Maxwell's  expressive  names,  Convergence  (or  Divergence)  and  Curl, 
have  sunk  into  the  very  heart  of  electromagnetic  theory."^  In  1913 
J.  B.  Shaw  generalized  the  Hamiltonian  v  for  space  of  n  dimensions, 
which  may  be  either  flat  or  curveo  Related  memoirs  are  due  to  G. 
Ricci  (1892),  T.  Levi-Civita  (1900),  H.  Maschke,  and  L.  Ingold  (1910). 
Quaternions  were  greatly  admired  in  England  from  the  start,  but  on 
the  Continent  they  received  less  attention.  P.  G.  Tait's  Elementary 
Treatise  hel{)ed  powerfully  to  spread  a  knowledge  of  them  in  England. 
A.  Cayley,  W.  K.  Clifford,  and  Tait  advanced  the  subject  somewhat 
by  original  contributions.  But  there  has  been  little  progress  in  recent 
years,  except  that  made  by  J.  J.  Sylvester  in  the  solution  of  quaternion 
equations,  nor  has  the  application  of  quaternions  to  physics  been  as 

'  C.  G.  Knott,  Life  and  Scientific  Work  of  Peter  GutJirie  Tail,  Cambridge,  191 1, 
PF).  143,  148. 

«  C.  G.  Knott,  op.  ciL,  p.  167. 


334  A  HISTORY  OF  MATHEMATICS 

extended  as  was  predicted.  The  change  in  notation  made  in  France 
by  Jules  Hoiiel  and  by  C,  A.  Laisant  has  been  considered  in  England 
as  a  wrong  step,  but  the  true  cause  for  the  lack  of  progress  is  perhaps 
more  deep-seated.  There  is  indeed  great  doubt  as  to  whether  the 
quaternionic  product  can  claim  a  necessary  and  fundamental  place 
in  a  system  of  vector  analysis.  Physicists  claim  that  there  is  a  loss 
of  naturalness  in  taking  the  square  of  a  vector  to  be  negative. 

Widely  different  opinions  have  been  expressed  on  the  value  of 
quaternions.  While  P.  G.  Tait  was  an  enthusiastic  champion  of  this 
science,  his  great  friend,  William  Thomson  (Lord  Kelvin),  declared 
that  they,  "though  beautifully  ingenious,  have  been  an  unmixed 
evil  to  those  who  have  touched  them  in  any  way,  including  Clerk 
Maxwell."  ^  A.  Cayley,  writing  to  Tait  in  1874,  said,  "I  admire  the 
equation  d(r=uqdpq~^  extremely— it  is  a  grand  example  of  the  pocket 
map."  Cayley  admitted  the  conciseness  of  quaternion  formulas,  but 
tliey  had  to  be  unfolded  into  Cartesian  form  before  they  could  be 
made  use  of  or  even  understood.  Cayley  wrote  a  paper  "On  Co-or- 
dinates versus  Quaternions"  in  the  Proceedings  of  the  Royal  Society 
of  Edinburgh,  Vol.  20,  to  which  Tait  replied  "On  the  Intrinsic  Nature 
of  the  Quaternion  Method." 

In  order  to  meet  more  adequately  the  wants  of  physicists,  /.  W. 
Gibbs  and  A.  Macfarlane  have  each  suggested  an  algebra  of  vectors 
with  a  new  notation.  Each  gives  a  definition  of  his  own  for  the 
product  of  two  vectors,  but  in  such  a  way  that  the  square  of  a  vector 
is  positive.  A  third  system  of  vector  analysis  has  been  used  by  Oliver 
Heaviside  in  his  electrical  researches. 

What  constitutes  the  most  desirable  notation  in  vector  analysis 
is  still  a  matter  of  dispute.  Chief,  among  the  various  suggestions,  are 
those  of  the  American  school,  started  by  J.  W.  Gibbs  and  those  of  the 
German-Italian  school.  The  cleavage  is  not  altogether  along  lines 
of  nationality.  L.  Prandl  of  Hanover  said  in  1904:  "After  long  delib- 
eration I  have  adopted  the  notation  of  Gibbs,  writing  a  .h  for  the 
inner  (scalar),  and  axb  for  the  outer  (vector)  product.  If  one  ob- 
serves the  rule  that  in  a  multiple  product  the  outer  product  must  be 
taken  before  the  inner,  the  inner  product  before  the  scalar,  then  one 
can  write  with  Gibbs  a  .bxc  and  ab.  c  without  giving  rise  to  doubt 
as  to  the  meaning."  ^ 

In  the  following  we  give  German-Italian  notations  first,  the  equiv- 
alent American  notation  (Gibbs')  second.  Inner  product  a  |  b,  a.  b; 
vector-product  ]  ab,axb;  alsoabc,  a. bxc;  ab  |  c,  (axb)xc;  ab  |  cd, 
(axb).(cx(i);'ab-,  (axb)-;  ab.cd,  (axb)x(cxd).  R.  Mehmke 
said  in  1904:  "The  notation  of  the  German-Italian  school  is  far  pref- 
erable to  that  of  Gibbs  not  only  in  logical  and  methodical,  but  also 
in  practical  respects." 

^  S.  P.  Thompson,  Life  of  Lord  Kelvin,  19 10,  p.  1138. 
2  Jahresh.  d.  d.  Math.  Vcreinig.,  Vol.  13,  p.  39. 


ALGEBRA  335 

In  1895  P.  Molenbroek  of  The  Hague  and  S.  Kimura,  then  at  Yale 
University,  took  the  first  steps  in  the  organization  of  an  International 
Association  for  Promoting  the  Study  of  Quaternions  and  Allied 
Systems  of  Mathematics.  P.  G.  Tait  was  elected  the  first  j^rcsident, 
])ut  could  not  accept  on  account  of  failing  health.  Alexander  Mac- 
farlane  (1851-1913)  of  the  University  of  Edinburgh,  later  of  the 
University  of  Texas  and  of  Lehigh  University,  served  as  secretary  of 
the  Association  and  was  its  president  at  the  time  of  his  death. 

At  the  international  congress  held  in  Rome  in  igoS  a  committee 
was  appointed  on  the  unification  of  vectorial  notations  but  at  the 
time  of  the  congress  held  in  Cambridge  in  191 2  no  definite  conclusions 
had  been  reached. 

Vectorial  notations  were  subjects  of  extended  discussion  in  UEn- 
seignement  mathemalique,  Vols.  11-14,  1909-1912,  between  C.  Burali- 
Forti  of  Turin,  R.  Marcolongo  of  Naples,  G.  Comberiac  of  Bourges, 
H.  C.  F.  Timerding  of  Strassburg,  F.  Klein  of  Gottingen,  E.  B. 
Wilson  of  Boston,  G.  Peano  of  Turin,  C.  G.  Knott  of  Edinburgh, 
Alexander  JVIacfarlane  of  Chatham  in  Canada,  E.  Carvallo  of  Paris, 
and  E.  Jahnke  of  Berlin.  In  America  the  relative  values  of  notations 
were  discussed  in  1916  by  E.  B.  Wilson  and  V.  C.  Poor. 

We  mention  two  topics  outside  of  ordinary  physics  in  which  vector 
analysis  has  figured.  The  generalization  of  A.  Einstein,  known  as 
the  principle  of  relativity,  and  its  interpretation  by  H.  Minkowski, 
have  opened  new  points  of  view.  Some  of  the  queer  consequences  of 
this  theory  disappear  when  kinematics  is  regarded  as  identical  with 
the  geometry  of  four-dimensional  space.  H.  Minkowski  and,  following 
him,  Max  Abraham,  used  vector  analysis  in  a  limited  degree,  Min- 
kowski usually  preferring  the  matrix  calculus  of  A.  Cayley.  A  more 
extended  use  of  vector  analysis  was  made  by  Gilbert  N.  Lewis  of  the 
University  of  California  who  introduced  in  his  extension  to  four  di- 
mensions some  of  the  original  features  of  H.  G.  Grassmann's  system. 

A  "dyname"  is,  according  to  J.  Pliicker  (and  others)  a  system  of 
forces  applied  to  a  rigid  body.  The  English  and  French  call  it  a 
"torsor."  In  1S99  this  subject  was  treated  by  the  Russian  A.  P. 
Kotjelnikoff  under  the  name  of  projective  theory  of  vectors.  In  1903 
E.  Study  of  Greifswald  brought  out  his  book,  Geometric  der  Dynamen, 
in  which  a  line-geometry  and  kinematics  are  elaborated,  partly  by 
the  use  of  group  theory,  which  are  carried  over  to  non-Euclidean 
spaces;  Study  claims  for  his  system  somewhat  greater  generality 
than  is  found  in  Hamilton's  quaternions  and  W.  K.  Clifford's  bi- 
quarternions. 

Hermann  Gunther  Grassmann  (1809-1877)  was  born  at  Stettin, 
attended  a  g}'mnasium  at  his  native  place  (where  his  father  was 
teacher  of  mathematics  and  physics),  and  studied  theology  in  Berlin 
for  three  years.  His  intellectual  interests  were  very  broad.  He  started 
as  a  theologian,  wrote  on  physics,  composed  texts  for  the  study  of 


336  A  HISTORY  OF  MATHEMATICS 

German,  Latin,  and  mathematics,  edited  a  political  paper  and  a  mis- 
sionary paper,  investigated  phonetic  laws,  wrote  a  dictionary  to  the 
Rig-Veda,  translated  the  Rig- Veda  in  verse,  harmonized  folk  songs 
in  three  voices,  carried  on  successfully  the  regular  work  of  a  teacher 
and  brought  up  nine  of  his  eleven  children — all  this  in  addition  to  the 
great  mathematical  creations  which  we  are  about  to  describe.  In 
1834  he  succeeded  J.  Steiner  as  teacher  of  mathematics  in  an  industrial 
school  in  Berlin,  but  returned  to  Stettin  in  1836  to  assume  the  duties 
of  teacher  of  mathematics,  the  sciences,  and  of  religion  in  a  school 
there. ^  Up  to  this  time  his  knowledge  of  mathematics  was  pretty 
much  confined  to  what  he  had  learned  from  his  father,  who  had 
^\Titten  two  books  on  "Raumlehre"  and  "Grossenlehre."  But  now 
he  made  his  acquaintance  with  the  works  of  S.  F.  Lacroix,  J.  L.  La- 
grange, and  P.  S.  Laplace.  He  noticed  that  Laplace's  results  could 
be  reached  in  a  shorter  way  by  some  new  ideas  advanced  in  his  father's 
books,  and  he  proceeded  to  elaborate;  this  abridged  method,  and  to 
apply  it  in  the  study  of  tides.  He  was  thus  led  to  a  new  geometric 
analysis.  In  1840  he  had  made  considerable  progress  in  its  develop- 
ment, but  a  new  book  of  Schleiermacher  drew  him  again  to  theology. 
In  1842  he  resumed  mathematical  research,  and  becoming  thoroughly 
convinced  of  the  importance  of  his  new  analysis,  decided  to  devote 
himself  to  it.  It  now  became  his  ambition  to  secure  a  mathematical 
chair  at  a  university,  but  in  this  he  never  succeeded.  In  1844  ap- 
peared his  great  classical  work,  the  Lineale  Ausdehnungslehre,  which 
was  full  of  new  and  strange  matter,  and  so  general,  abstract,  and  out 
of  fashion  in  its  mode  of  exposition,  that  it  could  hardly  have  had 
less  influence  on  European  mathematics  during  its  first  twenty  years, 
had  it  been  published  in  China.  K.  F.  Gauss,  J.  A.  Grunert,  and  A.  F. 
Mobius  glanced  over  it,  praised  it,  but  complained  of  the  strange 
terminology  and  its  "philosophische  Allgemeinheit."  Eight  years 
afterwards,  C.  A.  Bretschneider  of  Gotha  was  said  to  be  the  only 
man  who  had  read  it  through.  An  article  in  Crelle's  Journal,  in 
which  Grassmann  eclipsed  the  geometers  of  that  time  by  constructing, 
with  aid  of  his  method,  geometrically  any  algebraic  curve,  remained 
again  unnoticed.  Need  we  marvel  if  Grassmann  turned  his  attention 
to  other  subjects, — to  Schleiermacher's  philosophy,  to  politics,  to 
philology?  Still,  articles  by  him  continued  to  appear  in  Crelle's 
Journal,  and  in  1862  came  out  the  second  part  of  his  Ausdehnungslehre. 
It  was  intended  to  show  better  than  the  first  part  the  broad  scope  of 
the  Ausdehnungslehre,  by  considering  not  only  geometric  applica- 
tions, but  by  treating  also  of  algebraic  functions,  infinite  series,  and 
the  differential  and  integral  calculus.  But  the  second  part  was  no 
more  appreciated  than  the  first.  At  the  age  of  fifty-three,  this  won- 
derful man,  with  heavy  heart,  gave  uy)  mathematics,  and  directed  his 
energies  to  the  study  of  Sanskrit,  achieving  in  philology  results  which 
1  Victor  Schlegel,  Hermann  Grassmann,  Leipzig,  1878. 


ALGEBRA  337 

•/ere  better  appreciated,  and  which  vie  in  splendor  with  those  in 
^nathematics. 

Common  to  the  Ausdehnungslehre  and  to  quaternions  are  geometric 
addition,  the  function  of  two  vectors  represented  in  quaternions  by 
Sa^  and  Va0,  and  the  linear  vector  functions.  The  quaternion  is 
peculiar  to  W.  R.  Hamilton,  while  with  Grassmann  we  find  in  addition 
to  the  algebra  of  vectors  a  geometrical  algebra  of  wide  application, 
and  resembUng  A.  F.  ISIobius's  Barycentrische  Calcul,  in  which  the 
point  is  the  fundamental  element.  Grassmann  developed  the  idea 
of  the  "external  product,"  the  "internal  product,"  and  the  "open 
product."  The  last  we  now  call  a  matrix.  His  Ausdehnungslehre 
has  very  great  extension,  having  no  limitation  to  any  particular 
number  of  dimensions.  Only  in  recent  years  has  the  wonderful  rich- 
ness of  his  discoveries  begun  to  be  appreciated.  ^  second  edition  of 
the  Ausdehnungslehre  of  1844  was  printed  in  1877.  C.  S.  Peirce  gave 
a  representation  of  Grassmann's  system  in  the  logical  notation,  and 
E.  W.  Hyde  of  the  University  of  Cincinnati  ^\Tote  the  first  text-book 
on  Grassmann's  calculus  in  the  English  language. 

Discoveries  of  less  value,  wliich  in  part  covered  those  of  Grassmann 
and  Hamilton,  were  made  by  Barrc  de  Saint-Venant  (1797-1886), 
who  described  the  multiplication  of  vectors,  and  the  addition  of  vectors 
and  oriented  areas;  by  A.  L.  Caiichy,  whose  "clefs  algebriques"  were 
units  subject  to  combinatorial  multiplication,  and  were  applied  by 
the  author  to  the  theory  of  elimination  in  the  same  way  as  had  been 
done  earlier  by  Grassmann;  by  Giusto  Bellavitis  (1803-1S80),  Avho 
published  in  1835  and  1837  in  the  Annali  delle  Science  his  calculus  of 
aequipollences.  Bellavitis,  for  many  years  professor  at  Padua,  was 
a  self-taught  mathematician  of  much  power,  who  in  his  thirty-eighth 
year  laid  down  a  city  office  in  his  native  place,  Bassano,  that  he 
might  give  his  time  to  science. 

The  first  impression  of  H.  G.  Grassmann's  ideas  is  marked  in  the 
writings  of  II ei maun  Ilankcl,  who  published  in  1867  his  Vorlesungen 
iiber  die  Complexen  Zahlen.  Hankel,  then  docent  in  Leipzig,  had 
been  in  correspondence  ^\^th  Grassmann.  The  "alternate  numbers" 
of  Hankel  are  subject  to  his  law  of  combinatorial  multiplication.  In 
considering  the  foundations  of  Algebra  Hankel  affirms  the  principle  of 
the  permanence  of  formal  laws  previously  enunciated  incompletely 
by  G.  Peacock.  His  Complexe  Zahlen  was  at  first  little  read,  and  we 
must  turn  to  Victor  Schlegel  (1843-1905)  of  Hagen  as  the  successful 
interpreter  of  Grassmann.  Schlegel  was  at  one  time  a  young  col- 
league of  Grassmann  at  the  Marienstifts-Gymnasium  in  Stettin.  En- 
couraged by  R.  F.  A.  Clebsch,  Schlegel  wrote  a  System  der  Raimilehre^ 
T872-1875,  which  explained  the  essential  conceptions  and  operations 
of  the  Ausdehnungslehre. 

Grassmann's  ideas  spread  slowly.  In  1878  Clerk  Max\vell  wro'- 
j*^.  G.  Tait:  "Do  you  know  Grassmann's  Ausdehnungslehre?    Spottis 


33^ 


A  HISTORY  OF  MATHEMATICS 


woode  spoke  of  it  in  Dublin  as  something  ab'^ve  and  beyond  4nions. 
I  have  not  seen  it,  but  Sir  W.  Hamilton  of  Edinburgh  used  to  say 
that  the  greater  the  extension  the  smaller  the  intention." 

Multiple  algebra  was  powerfully  advanced  by  B.  Peirce,  whose 
theory  is  not  geometrical,  as  are  those  of  W.  R.  Hamilton  and  H.  G. 
Grassmann.  Benjamin  Peirce  (1S09-18S0)  was  born  at  Salem,  Mass., 
and  graduated  at  Harvard  College,  having  as  undergraduate  carried 
the  study  of  mathematics  far  beyond  the  limits  of  the  college  course. 
When  N.  Bowditch  was  preparing  his  translation  and  commentary 
of  the  Mecanique  Celeste,  young  Peirce  helped  in  reading  the  proof- 
sheets.  He  was  made  professor  at  Elarvard  in  1833,  a  position  which 
he  retained  until  his  death.  For  some  years  he  was  in  charge  of  the 
Nautical  Ahnanac  and  superintendent  of  the  United  States  Coast 
Survey.  He  published  a  series  of  college  text-books  on  mathematics, 
an  Analytical  Mechanics,  1855,  and  calculated,  together  with  Sears 
C.  Walker  of  Washington,  the  orbit  of  Neptune.  Profound  are  his 
researches  on  Linear  Associative  Algebra.  The  first  of  several  papers 
thereon  was  read  at  the  first  meeting  of  the  American  Association 
for  the  Advancement  of  Science  in  1864.  Lithographed  copies  of  a 
memoir  were  distributed  among  friends  in  1870,  but  so  small  seemed 
to  be  the  interest  taken  in  this  subject  that  the  memoir  was  not 
printed  until  1881  {Am.  Jour.  Math.,  Vol.  IV,  No.  2).  Peirce  works 
out  the  multiplication  tables,  first  of  single  algebras,  then  of  double 
algebras,  and  so  on  up  to  sextuple,  making  in  all  162  algebras,  which 
he  shows  to  be  possible  on  the  consideration  of  symbols  A,  B,  etc., 
which  are  linear  functions  of  a  determinate  number  of  letters  or  units 
i,  j,  k,  I,  etc.,  with  coefficients  which  are  ordinary  analytical  magni- 
tudes, real  or  imaginary, — the  letters  i,  j,  etc.,  being  such  that  every 
binary  combination  i^,  ij,ji,  etc.,  is  equal  to  a  Hnear  function  of  the 
letters,  but  under  the  restriction  of  satisfying  the  associative  law.-^ 
Charles  S.  Peirce,  a  son  of  Benjamin  Peirce,  and  one  of  the  foremost 
writers  on  mathematical  logic,  showed  that  these  algebras  were  all 
defective  forms  of  quadrate  algebras  which  he  had  previously  dis- 
covered by  logical  analysis,  and  for  which  he  had  devised  a  simple 
notation.  Of  these  quadrate  algebras  quaternions  is  a  simple  example; 
nonions  is  another.  C.  S.  Peirce  showed  that  of  all  linear  associative 
algebras  there  are  only  three  in  which  division  is  unambiguous.  These 
are  ordinary  single  algebra,  ordinary  double  algebra,  and  quaternions, 
from  which  the  imaginary  scalar  is  excluded.  He  showed  that  his 
father's  algebras  are  operational  and  matricular.  Lectures  on  multiple 
algebra  were  delivered  by  J.  J.  Sylvester  at  the  Johns  Hopkins  Uni- 
versity, and  published  in  various  journals.  They  treat  largely  of  the 
algebra  of  matrices. 

While  Benjamin  Peirce's  comparative  anatomy  of  linear  algebras 
was  favorably  received  in  England,  it  was  criticised  in  Germany  as 
1  A.  Cayley,  Address  before  British  Association,  1883. 


ALGEBRA  339 

being  vague  and  based  on  arbitrary  principles  of  classification.  Ger- 
man writers  along  this  line  are  Eduard  Study  and  Georg  W.  Scheffers. 
An  estimate  of  B.  Peirce's  linear  associative  algebra  was  given  in 
igo2  by  H.  E.  Hawkes/  who  extends  Peirce's  method  and  shows  its 
full  power.  In  iSqS  Ehe  Cartan  of  the  University  of  Lyon  used  the 
characteristic  equation  to  develop  several  general  theorems;  he  ex- 
hibits the  semi-simple,  or  Dedekind,  and  the  pseudo-nut,  or  nilpotent, 
sub-algebras;  he  shows  that  the  structure  of  every  algebra  may  be 
represented  by  the  use  of  double  units,  the  first  factor  being  quad- 
rate, the  second  non-quadrate.  Extensions  of  B.  Peirce's  results  were 
made  also  by  Henry  Taber.  Olive  C.  Hazlett  gave  a  classification  of 
nilpotent  alegbras. 

As  shown  above,  C.  S.  Peirce  advanced  this  algebra  by  using  the 
matrix  theory.  Papers  along  this  fine  are  due  to  F.  G.  Frobenius  and 
J.  B.  Shaw.  The  latter  "shows  that  the  equation  of  an  algebra  de- 
termines its  quadrate  units,  and  certain  of  the  direct  units;  that  the 
other  units  form  a  nilpotent  system  which  with  the  quadrates  may 
be  reduced  to  certain  canonical  forms.  The  algebra  is  thus  made  a 
sub-algebra  under  the  algebra  of  the  associative  units  used  in  these 
canonical  forms.  Frobenius  proves  that  every  algebra  has  a  Dede- 
kind sub-algebra,  whose  equation  contains  all  factors  in  the  equation 
of  the  algebra.  This  is  the  semi-simple  algebra  of  Cartan.  He  also 
showed  that  the  remaining  units  form  a  nilpotent  algebra  whose  units 
may  be  regularized"  (J.  B.  Shaw).  More  recently,  J.  B.  Shaw  has 
extended  the  general  theorems  of  linear  associative  algebras  to  such 
algebras  as  have  an  infinite  number  of  units. 

Besides  the  matrix  theory,  the  theory  of  continuous  groups  has  been 
used  in  the  study  of  linear  associative  algebra.  This  isomorphism 
was  first  pointed  out  by  H.  Poincare  (18S4);  the  method  was  followed 
by  Georg  W.  Scheffers  who  classified  algebras  as  quatemionic  and 
non-quaternionic  and  worked  out  complete  lists  of  all  algebras  to 
order  five.  Theodor  Molien,  in  1S93,  then  in  Dorpat,  demonstrated 
"that  quaternionic  algebras  contain  independent  quadrates,  and  that 
quaternionic  algebras  can  be  classified  according  to  non-quaternionic 
types  "  (J.  B.  Shaw).  An  elementary  exposition  of  the  relation  between 
linear  algebras  and  continuous  groups  was  given  by  L.  E.  Dickson  "  of 
Chicago.  This  relation  "enables  us  to  translate  the  concepts  and 
theorems  of  the  one  subject  into  the  language  of  the  other  subject. 
It  not  only  doubles  our  total  knowledge,  but  gives  us  a  better  insight 
into  either  subject  by  exhibiting  it  from  a  new  point  of  view."  The 
theory  of  matrices  was  developed  as  early  as  1858  by  A.  Cayley  in  an 
important  memoir  which,  in  the  opinion  of  J.  J.  Sylvester,  ushered  in 

'  H.  E.  Hawkes  in  Am.  Jour.  Math.,  Vol.  24,  1902,  p.  87.  We  are  using  also 
J.  B.  Shaw's  Synopsis  of  Linear  Associative  Algebra,  Washington,  D.  C,  1907. 
Introduction.    Shaw  gives  bibliography. 

2  Bull.  Am.  Math.  Soc,  Vol.  22,  1915,  p.  53. 


340  A  HISTORY  OF  MATHEMATICS 

the  reign  of  Algebra  the  Second.  W.  K.  Chfford,  Sylvester,  H.  Taber, 
C.  II.  Chapman,  carried  the  investigations  much  further.  The  origi- 
nator of  matrices  is  really  W.  R.  Hamilton,  but  his  theory,  pubhshed 
in  his  Lectures  on  Quaternions,  is  less  general  than  that  of  Cayley. 
The  latter  makes  no  reference  to  Hamilton. 

The  theory  of  determinants  ^  was  studied  by  Hene  Wronski  (1778- 
1S53),  a  poor  Polish  enthusiast,  living  most  the  time  in  France,  whose 
egotism  and  wearisome  style  tended  to  attract  few  followers,  but  who 
made  some  incisive  criticisms  bearing  on  the  philosophy  of  mathe- 
matics.^ He  studied  four  special  forms  of  determinants,  which  were 
extended  by  Ileinrkh  Ferdinand  Scherk  (1798-1885)  of  Bremen  and 
Ferdinand  Schweins  (1780-1856)  of  Heidelberg.  In  1S38  Liouville 
demonstrated  a  property  of  the  special  forms  which  were  called 
"wronskians"  by  Thomas  Muir  in  1881.  Determinants  received  the 
attention  of  Jacques  P.  M.  Binet  (i  786-1 856)  of  Paris,  but  the  great 
master  of  this  subject  was  A.  L.  Cauchy.  In  a  paper  {Jour,  de  Vecole 
Polyt.,  IX.,  16)  Cauchy  developed  several  general  theorems.  He  in- 
troduced the  name  determinant,  a  term  used  by  K.  F.  Gauss  in  1801  in 
the  functions  considered  by  him.  In  1826  C.  G.  J.  Jacobi  began  using 
this  calculus,  and  he  gave  brilliant  proof  of  its  power.  In  1841  he 
wrote  extended  memoirs  on  determinants  in  Crelle's  Journal,  which 
rendered  the  theory  easily  accessible.  In  England  the  study  of  linear 
transformations  of  quantics  gave  a  powerful  impulse.  A.  Cayley  de- 
veloped skew-determinants  and  Pfaflians,  and  introduced  the  use  of 
determinant  brackets,  or  the  familiar  pair  of  upright  lines.  The  more 
general  consideration  of  determinants  whose  elements  are  formed  from 
the  elements  of  given  determinants  was  taken  up  by  J.  J.  Sylvester 
(1S51)  and  especially  by  L.  Kronecker  who  gave  an  elegant  theorem 
known  by  his  name.^  Orthogonal  determinants  received  the  atten- 
tion of  A.  Cayley  in  1846,  in  the  study  of  ;i"  elements  related  to  each 
other  by  ln{n+i)  equations,  also  of  L.  Kronecker,  F.  Brioschi  and 
others.  Maximal  values  of  determinants  received  the  attention  of 
J.  J.  Sylvester  (1867),  and  especially  of  J.  Hadamard  (1893)  who 
proved  that  the  square  of  a  determinant  is  never  greater  than  the 
norm-product  of  the  lines. 

Anton  Puchta  (1851-1903)  of  Czernowitz  in  1878  and  M.  Noether 
in  18S0  showed  that  a  symmetric  determinant  may  be  expressed  as 
the  product  of  a  certain  number  of  factors,  linear  in  the  elements. 
Determinants  which  are  formed  from  the  minors  of  a  determinant 
were  investigated  by  J.  J.  Sylvester  in  185 1,  to  whom  we  owe  the 

'  Thomas  IMuir,  The  Theory  of  Determinants  in  the  Historical  Order  of  Develop- 
imnt  [Vol.  I],  2nd  Ed.,  London,  1906;  Vol.  II,  Period  1841  to  1S60,  London,  1911. 
il-  ir  was  Superintendent-General  of  Education  in  Cape  Colony. 

-On  Wronski,  see  J.  Bertrand  in  Journal  dcs  Savants,  1807,  and  in  Revue  des 
Dciix-Mondcs,  Feb.,  1897.  See  also  L'InlcrmeJiaire  des  Malhetnaticiens,  Vol.  23, 
1916,  pp.  113,  1O4-167,  1S1-103. 

'  E.  Pascal,  Die.  Delerminanlcn ,  trans!,  by  II.  Leitzmann,  1900,  p.  107. 


ALGEBRA  341 

'umbral  notation,"  by  W.  fpottiswoode  in  1S56,  and  later  by  G. 
Janni,  AI.  Reiss  (1S05-1S69),  E.  d'Ovidio,  II.  Picquet,  E.  Hunyadi 
(1S3S-1S89)  E.  Barbier,  C.  A.  Van  Velzer,  E.  Nelto,  G.  Frobenius, 
and  others.  Many  researches  on  determinants  appertain  to  special 
forms.  "Continuants"  are  due  to  J.  J.  Sylvester;  "alternants,"  origi- 
nated by  A.  L.  Cauchy,  have  been  developed  by  C.  G.  J.  Jacobi, 
Nicolo  Trudi  (1S11-1884)  of  Naples,  H.  Nagelsbach,  and  G.  Garbieri; 
"axisymmetric  d:terminants,"  first  used  by  Jacobi,  have  been  studied 
by  V.  A.  Lebesgue,  J.  J.  Sylvester,  and  L.  O.  Hesse;  "circulants"  are 
du  to  Eugene  Charles  Catalan  (1814-1894)  of  Liose,  William  Spottis- 
woode  (1S25-1883)  of  Oxford,  J.  Vv.  L.  Glaisher^  and  R.  F.  Scott; 
for  "centro-symmetric  determinants"  we  are  indebted  to  G.  Zehfuss 
of  Heidelberg.  V.  Nachreiner  and  S.  Giinther,  both  of  Munich, 
pointed  out  relations  between  determinants  and  continued  fractions; 
R.  F.  Scott  uses  H.  Hankel's  alternate  numbers  in  his  treatise.  A 
class  of  determinants  which  have  the  same  importance  in  linear  inte- 
gral equations  as  do  ordinary  determinants  for  linear  equations  with 
n  unknowns  was  worked  out  by  E.  Fredholm  {Acta  math.,  1903)  and 
again  by  D.  Hilbert  who  reaches  them  as  limiting  expressions  of  or- 
dinary determinants. 

An  achievement  of  considerable  significance  was  the  introduction 
in  i860  of  infinite  determinants  by  Eduard  Fiirstcnan  in  a  method 
of  approximation  to  the  roots  of  algebraic  equations.  Determinants  of 
an  infmite  order  were  used  by  Thcodor  Kotteritzsch  of  Grimma  in 
Saxony,  in  two  papers  on  the  solution  of  an  infinite  system  of  linear 
equations  {Zeitsch.f.  Math.  u.  Physik,  Vol.  14,  1870).  Independently, 
infinite  determinants  were  introduced  in  1877  by  George  William 
Hill  of  Washington  in  an  astronomical  paper  (Collected  Works,  Vol. 
I,  1905,  p.  243).  In  1S84  and  1885  H.  Poincarc  called  attention  to 
these  determinants  as  developed  by  Hill  and  investigated  them  further. 
Their  theory  was  elaborated  later  by  Helge  von  Koch  and  Erhard 
Schmidt  (1908). 

In  recent  years  the  theory  of  the  solution  of  a  system  of  linear 
equations  has  been  presented  in  an  elegant  form  by  means  of  what 
is  known  as  the  rank  of  a  determinant.  In  particular,  G.  A.  Miller 
has  thus  developed  a  necessary  and  sufficient  condition  that  a  given 
unknown  in  a  consistent  system  of  linear  equations  have  only  one 
value  while  some  of  the  other  unknowns  may  assume  an  infinite 
number  of  values.^ 

Text-books  on  determinants  were  written  by  W.  Spottiswoode 
(1851),  F.  Brioschi  (1854),  R.  Baltzer  (1S57),  S.  Giinther  (1875), 
G.  J.  Dostor  (1877),  R.  F.  Scott  (1880),  T.  Muir  (1882),  P.  H.  Hanus 
(1886),  G.  W.  H.  Kowalewski  (1909). 

The  symbol  n!  for  "factorial  n,"  now  universally  used  in  algebra, 
is  due  to  Christian  Kramp  (1760-1826)  of  Strassburg.  who  used  it  in 
'  Atn.  Math.  Monthly,  Vol.  17,  loio.  r>   -u 


342  A  HISTORY  OF  MATHEMATICS 

1808.  The  symbol  =  to  express  identity  was  first  used  by  G.  F.  B. 
Riemann.^ 

Modern  higher  algebra  is  especially  occupied  with  the  theory  of 
linear  transformations.  Its  development  is  mainly  the  work  of  A. 
Cayley  and  J.  J.  Sylvester. 

Arthur  Cayley  (1821-1895),  born  at  Richmond,  in  Surrey,  was 
educated  at  Trinity  College,  Cambridge.  He  came  out  Senior  Wrang- 
ler in  1842.  He  then  devoted  some  years  to  the  study  and  practice 
of  law.  While  a  student  at  the  bar  he  went  to  Dublin  and,  alongside 
of  G.  Salmon,  heard  W.  R.  Hamilton's  lectures  on  quaternions.  On 
the  foundation  of  the  Sadlerian  professorship  at  Cambridge,  he  ac- 
cepted the  ofifer  of  that  chair,  thus  giving  up  a  profession  promising 
wealth  for  a  very  modest  provision,  but  which  would  enable  him  to  give 
all  his  time  to  mathematics.  Cayley  began  his  mathematical  publica- 
tions in  the  Cambridge  Mathematical  Journal  while  he  was  still  aa 
undergraduate.  Some  of  his  most  brilliant  discoveries  were  maoe 
during  the  time  of  his  legal  practice.  There  is  hardly  any  subject 
in  pure  mathematics  which  the  genius  of  Cayley  has  not  enriched,  but 
most  important  is  his  creation  of  a  new  branch  of  analysis  by  his 
theory  of  invariants.  Germs  of  the  principle  of  invariants  are  found 
in  the  writings  of  J.  L.  Lagrange,  K.  F.  Gauss,  and  particularly  of 
G.  Boole,  who  showed,  in  1841,  that  invariance  is  a  property  of  dis- 
criminants generally,  and  who  applied  it  to  the  theory  of  orthogonal 
substitution.  Cayley  set  himself  the  problem  to  determine  a  priori 
what  functions  of  the  coefficients  of  a  given  equation  possess  this  prop- 
erty of  invariance,  and  found,  to  begin  with,  in  1845,  that  the  so- 
called  "hyper-determinants"  possessed  it.  G.  Boole  made  a  number 
of  additional  discoveries.  Then  J.  J.  Sylvester  began  his  papers  in  the 
Cambridge  and  Dublin  Mathematical  Journal  on  the  Calculus  of  Forms. 
After  this,  discoveries  followed  in  rapid  succession.  At  that  time  Cay- 
ley and  Sylvester  were  both  residents  of  London,  and  they  stimulated 
each  other  by  frequent  oral  communications.  It  has  often  been  dif- 
ficult to  determine  how  much  really  belongs  to  each.  In  18S2,  when 
Sylvester  was  professor  at  the  Johns  Hopkins  University,  Cayley 
lectured  there  on  Abelian  and  theta  functions. 

Of  interest  is  Cayley's  method  of  work.  A.  R.  Forsyth  describes  it 
thus:  "When  Cayley  had  reached  his  most  advanced  generalizations 
he  proceeded  to  establish  them  directly  by  some  method  or  other, 
though  he  seldom  gave  the  clue  by  which  they  had  first  been  obtained: 
a  proceeding  which  does  not  tend  to  make  his  papers  easy  reading.  .  .  . 
His  literary  style  is  direct,  simple  and  clear.  His  legal  training  had 
an  influence,  not  merely  upon  his  mode  of  arrangement  but  also  upon 
his  expression;  the  result  is  that  his  papers  are  severe  and  present  a 
curious  contrast  to  the  luxuriant  enthusiasm  which  perv^ades  so  many 

^  L.  Kronecker,  Vcyiicsiingen  iihcr  Zahlentheorie,  igoi,  p.  86 


ALGEBRA  343 

of  Sylvester's  papers."  ^  Curiously,  Cayley  took  little  interest  in 
quaternions. 

James  Joseph  Sylvester  (1814-1897)  was  born  in  London.  His 
father's  name  Avas  Abraham  Josejih;  his  eldest  brother  assumed  in 
America  the  name  of  Sylvester,  and  he  adojoted  this  name  too.  About 
the  age  of  16  he  was  awarded  a  prize  of  S500  for  solving  a  question 
in  arrangements  for  contractors  of  lotteries  in  the  United  States.^ 
In  183 1  he  entered  St.  John's  College,  Cambridge,  and  came  out 
Second  Wrangler  in  1837,  George  Green  being  fourth.  Sylvester's 
Jewish  origin  incapacitated  him  from  taking  a  degree.  From  1838  to 
1840  he  was  professor  of  natural  philosophy  at  what  is  now  University 
College,  London;  in  1841  he  became  professor  of  mathematics  at  the 
University  of  Virginia.  In  a  quarrel  with  two  of  his  students  he  slightly 
wounded  one  of  them  with  a  metal  pointed  cane,  whereupon  he  re- 
turned hurriedly,  to  England.  In  1844  he  served  as  an  actuar}^;  in 
1846  he  became  a  student  at  the  Inner  Temple  and  was  called  to  the 
bar  in  1850.  In  1S46  he  became  associated  with  A.  Cayley;  often 
they  walked  round  the  Courts  of  Lincoln's  Inn,  perhaps  discussing 
the  theory  of  invariants,  and  Cayley  (says  Sylvester)  "habitually 
discoursing  pearls  and  rubies."  Sylvester  resumed  mathematical 
research.  He,  Cayley  and  William  Rowan  Hamilton  entered  upon 
discoveries  in  pure  mathematics  that  are  unequalled  in  Great  Britain 
since  the  time  of  I.  Newton.  Sylvester  made  the  friendship  of  G.  Sal- 
mon whose  books  contributed  greatly  to  bring  the  results  of  Cayley 
and  Sylvester  within  easier  reach  of  the  mathematical  public.  From 
1855  to  1870  Sylvester  was  professor  at  the  Royal  Military  Academy 
at  Woolwich,  but  showed  no  great  efficiency  as  an  elementary  teacher. 
There  are  stories  of  his  housekeeper  pursuing  him  from  home  carrying 
his  collar  and  necktie.  From  1876  to  1883,  he  was  professor  at  the 
Johns  Hopkins  University,  where  he  was  happy  in  being  free  to  teach 
whatever  he  wished  in  the  way  he  thought  best.  He  became  the  first 
editor  of  the  American  Journal  of  Mathematics  in  1878.  In  1884  he 
was  elected  to  succeed  H.  J.  S.  Smith  in  the  chair  of  Savilian  professor 
of  geometry  at  Oxford,  a  chair  once  occupied  by  Henry  Briggs,  John 
WaUis  and  Edmund  Halley. 

Sylvester  sometimes  amused  himself  writing  poetry.  His  Laws  of 
Verse  is  a  curious  booklet.  At  the  reading,  at  the  Peabody  Institute 
in  Baltimore,  of  his  Rosalind  poem,  consisting  of  about  400  lines  all 
rhyming  with  "Rosalind,"  he  first  read  all  his  explanator}'  footnotes, 
so  as  not  to  interrupt  the  poem;  these  took  one  hour  and  a-half.  Then 
he  read  the  poem  itself  to  the  remnant  of  his  audience. 

^  Proceed.  London  Royal  Society,  Vol.  58,  1895,  pp.  23,  24. 

2  H.  F.  Baker's  Bioffraphical  Notice  in  The  Collected  Math.  Papers  of  J.  J.  Syl- 
vester, Vol.  IV,  Cambridge,  19 12.  We  have  used  also  P.  A.  MacMahon's  notice  in 
Proceed.  Royal  Soc.  of  Lciidon,  Vol.  63,  1898,  p.  ix.  For  Sylvester's  activities  in 
Baltimore,  see  Fabian  Franklin  in  Johns  Hopkins  Univ.  Circulars,  June,  1807;  F. 


344  A  HISTORY  OF  MATHEMATICS 

Sylvester's  first  papers  were  on  Fresnel's  optic  theory,  1837.  Two 
years  later  he  wrote  on  C.  Sturm's  memorable  theorem.  Sturm 
once  told  him  that  the  theorem  originated  in  the  theory  of  the  com- 
pound pendulum.  Stimulated  by  A.  Cayley  he  made  important  in- 
vestigations on  modern  algebra.  He  wrote  on  elimination,  on  trans- 
formation and  canonical  forms,  in  which  the  expression  of  a  cubic 
surface  by  five  cubes  is  given,  on  the  relation  between  the  minor  de- 
terminants of  linearly  equivalent  quadratic  functions,  in  which  the 
notion  of  invariant  factors  is  implicit,  while  in  1S52  appeared  the 
first  of  his  papers  on  the  principles  of  the  calculus  of  forms.  In  a 
reply  that  he  made  in  1869  to  Huxley  who  had  claimed  that  mathe- 
matics was  a  science  that  knows  nothing  of  observation,  induction, 
invention  and  experimental  verification,  Sylvester  narrated  his  per- 
sonal experience:  "I  discovered  and  developed  the  whole  theory  of 
canonical  binary  forms  for  odd  degrees,  and,  as  far  as  yet  made  out, 
for  even  degrees  too,  at  one  evening  sitting,  with  a  decanter  of  port 
wine  to  sustain  nature's  flagging  energies,  in  a  back  office  in  Lincoln's 
Inn  Fields.  The  work  was  done,  and  well  done,  but  at  the  usual 
cost  of  racking  thought — a  brain  on  fire,  and  feet  feeling,  or  feelingless, 
as  if  plunged  in  an  ice-pail.  That  night  we  slept  no  more."  His  reply 
to  Huxley  is  interesting  reading  and  bears  strongly  on  the  qualities 
of  mental  activity  involved  in  mathematical  research.  In  1859  he 
gave  lectures  on  partitions,  not  published  until  1897.  He  wrote  on 
partitions  again  in  Baltimore.  In  1S64  followed  his  famous  proof 
of  Newton's  rule.  A  certain  fundamental  theorem  in  invariants 
which  had  formed  the  basis  of  an  important  section  of  A.  Cayley's 
work,  but  had  resisted  proof  for  a  quarter  of  a  century  was  demon- 
strated by  Sylvester  in  Baltimore.  Noteworthy  are  his  memoirs  on 
Chebichev's  method  concerning  the  totality  of  prime  numbers  within 
certain  limits,  and  his  latent  roots  of  matrices.  His  researches  on 
invariants,  theory  of  equations,  multiple  algebra,  theory  of  numbers, 
linkages,  probability,  constitute  important  contributions  to  mathe- 
matics. His  final  studies,  entered  upon  after  his  return  to  Oxford, 
were  on  reciprocants  or  functions  of  differential  coefficients  whose 
form  is  unaltered  by  certain  linear  transformations  of  the  variables, 
and  a  generahzation  of  the  theory  of  concomitants.  In  191 1,  G. 
Greenhill  told  reminiscently  ^  how  Sylvester  got  everybody  interested 
in  reciprocants,  "now  clean  forgotten";  "One  day  after,  Sylvester 
was  noticed  walking  alone,  addressing  the  sky,  asking  it:  'Are  Recipro- 
cants Bosh?  Berry  of  King's  says  the  Reciprocant  is  all  Bosh!' 
There  was  no  reply,  and  Sylvester  himself  was  tiring  of  the  subject, 
and  so  Berry  escaped  a  castigation.  But  recently  I  had  occasion 
from  the  Aeronautical  point  of  view  to  work  out  the  theory  of  a  Vortex 

Cajori,  Teaching;,  and  History  of  Mathematics  in  tJic  United  Stales,  Wash'ngton,  1890, 
pn.  26i-.^72. 

'  Mathematical  Gazette,  Vol.  6,  19 12,  p.  108. 


ALGEBRA 


345 


inside  a  Polygon,  an  eddy  whirlwind  such  as  Chavez  had  lo  encounter 
in  the  angle  of  the  precipices,  flying  over  the  Simplon  Pass.  The 
analysis  in  some  cases  seemed  strangely  familiar,  and  at  last  I  recog- 
nized the  familiar  Reciprocant.  .  .  .  Difference  in  Similarity  and 
Similarity  in  Difference  has  been  called  the  motto  of  our  science."  ^ 

In  the  American  Journal  of  Mathematics  are  memoirs  on  binary  and 
ternary  quantics,  elaborated  partly  with  aid  of  F.  Franklin,  then 
professor  at  the  Johns  Hopkins  University.  The  theory  of  reciprocants 
is  more  general  than  one  on  differential  invariants  by  G.  H.  Halphen 
(1878),  and  has  been  developed  further  by  J.  Hammond  of  Cambridge, 
P.  A.  McIMahon  of  Woolwich,  A.  R.  Forsyth  now  of  London,  and 
others.  Sylvester  playfully  lays  claim  to  the  appellation  of  the  Mathe- 
matical Adam,  for  the  many  names  he  has  introduced  into  mathe- 
matics. Thus  the  terms  invariant,  discriminant,  Hessian,  Jacobian, 
are  his.  That  not  only  elementar}'  pupils,  but  highly  trained  math- 
ematicians as  well,  may  be  attracted  or  repelled  by  the  kind  of  symbols 
used,  is  illustrated  by  the  experience  of  K.  Weierstrass  who  related 
that  he  followed  Sylvester's  papers  on  the  theory  of  algebraic  forms 
very  attentively  until  Sylvester  began  to  employ  Hebrew  characters. 
That  was  more  than  he  could  stand  and  after  that  he  quit  him.^ 

The  great  theory  of  invariants,  developed  in  England  mainly  by 
A.  Cayley  and  J.  J.  Svlvester,  came  to  be  studied  earnestly  in  Ger- 
many, France,  and  Italy.  Ch.  Hermite  discovered  evectants,  and 
the  theorem  of  reciprocity  named  after  him,  whereby  "to  every  co vari- 
ant of  degree  n  in  the  coefficients  of  the  quantic  of  order  m,  there 
corresponds  a  covariant  of  degree  wz  in  the  coefficients  of  a  quantic  of 
order  n.  He  discovered  the  skew  invariant  of  the  quintic,  which  was 
the  first  example  of  any  skew  invariant.  He  discovered  the  linear  co- 
variants  belonging  to  quantics  of  odd  order  greater  than  3,  and  he  ap- 
plied them  to  obtain  the  typical  expression  of  the  quantic  in  which  the 
coefficients  are  invariants.  He  also  invented  the  associated  covariants 
of  a  quantic;  these  constitute  the  simplest  set  of  algebraically  complete 
systems  as  distinguished  from  systems  that  are  linearly  complete."  ^ 

In  Italy,  F.  Brioschi  of  JMilan  and  Fad  de  Bruno  (1S25-1888)  con- 
tributed to  the  theory  of  invariants,  the  latter  writing  a  text-book 
on  binary  forms  (1S76),  which  ranks  by  the  side  of  G.  Salmon's  treatise 
and  those  of  R.  F.  A.  Clebsch  and  P.  Gordan. 

Francesco  Brioschi  (1824-1S97)  in  1852  became  professor  of  applied 
mathematics  at  the  University  of  Pavia  and  in  1S62  was  commissioned 
by  the  government  to  organize  the  Instituto  tecnico  superiore  at 
IMilan,  where  he  filled  until  his  death  the  chair  of  hydraulics  and 

^  Mathematical  Gazette,  Vol.  6,  191 2,  p.  108. 

^  E.  Lampe  (i840-»9iS),  in  Naturunssenschaftliche  Rundschau,  Bd.  12,  1897, 
p.  361;  quoted  by  R.  E.  Moritz,  Memorabilia  Malhcmalica,  1914,  p.  180. 

^  Proceed,  of  the  Roy.  Soc.  of  London,  Vol.  45,  1905,  p.  144.  Obituary  notice  of 
Hermite  by  A.  R.  Forsj'th. 


346  A  HISTORY  OF  MATHEMATICS 

analysis.  With  Abbe  Barnaba  TortoUni  (1S08-1S74)  he  founded  in 
1S58  the  Aitiiali  di  matematica  pura  ed  applicata.  Among  his  pupils 
at  Pavia  were  L.  Cremona  and  E.  Beltrami.  V.  Volterra  narrates  ^ 
how  F.  Brioschi  in  1858,  with  two  other  young  Italians,  Enrico  Betti 
(1823-1892),  later  professor  at  the  University  of  Pisa,  and  Felice 
Casorati  (1835-1S90),  later  professor  at  the  University  of  Pavia, 
started  on  a  journey  to  enter  into  relations  with  the  foremost  mathe- 
maticians of  France  and  Germany.  "The  scientific  existence  of  Italy 
as  a  nation"  dates  from  this  journey.  "It  is  to  the  teaching,  labors, 
and  devotion  of  these  three,  to  their  influence  in  the  organization  of 
advanced  studies,  to  the  friendly  scientific  relations  that  they  insti- 
tuted between  Italy  and  foreign  countries,  that  the  existence  of  a 
school  of  analysts  in  Italy  is  du:\" 

In  Gennany  the  early  theory  of  invariants,  as  developed  by  Cayley, 
Sylvester,  and  Salmon  in  England,  Hermite  in  France  and  F.  Brioschi 
in  Italy,  did  not  draw  attention  until  1858  when  Siegfried  Heinrich 
Aronhold  (1819-1884)  of  the  technical  high  school  in  BerHn  pointed 
out  that  Hesse's  theory  of  ternary  cubic  forms  of  1844  involved  in- 
variants by  which  that  theory  could  be  rounded  out.  F.  G.  Eisenstein 
and  J.  Steiner  had  also  given  early  publication  to  isolated  develop- 
ments involving  the  invariantal  idea.  In  1863  Aronhold  gave  a 
systematic  and  general  exposition  of  invariant  theory  (Crelle,  62). 
He  and  Clebsch  used  a  notation  of  their  own,  the  symbolical  notation, 
different  from  Cayley's,  which  was  used  in  the  further  developments 
of  the  theory  in  Germany.  Great  developments  were  started  about 
1868,  when  R.  F.  A.  Clebsch  and  P.  Gordan  wrote  on  types  of  binary 
forms,  L.  Kroneckcr  and  E.  B.  ChristofTel  on  bilinear  forms,  F.  Klein 
and  S.  Lie  on  the  invariant  theory  connected  with  any  group  of  linear 
substitutions.  Paul  Gordan  (1837-1912)  was  born  at  Erlangen  and 
became  professor  there.  He  produced  papers  on  finite  groups,  par- 
ticularly on  the  simple  group  of  order  168  and  its  associated  curve 
y^z+z^x+x^y=o.  His  best  known  achievement  is  the  proof  of  the 
existence  of  a  com.plete  system  of  concomitants  for  any  given  binary 
form.-  While  Clebsch  aimed  in  his  researches  to  devise  methods  by 
which  he  could  study  the  relationships  between  invariantal  forms 
(Formcnverwandtschaft),  the  chief  aim  of  Aronhold  was  to  examine 
the  equivalence  or  the  linear  transformation  of  one  form  into  another.^ 
Investigations  along  this  line  are  due  to  E.  B.  Christoffel,  who  showed 
that  the  number  of  arbitrary  parameters  contained  in  the  substitution 
coefficients  equals  that  of  the  absolute  invariants  of  the  form,  K. 
Weierstrass  who  gave  a  general  treatment  of  the  equivalence  of  two 

'  Bull.  Am-.  Math.  Soc,  Vol.  7,  1900,  p.  60. 

'^Nature,  Vcl.  90,  1913,  p.  597. 

^  We  are  using  Franz  Meyer,  "  Kericht  iibjr  dc-n  gegenwartisr^n  Stand  dor  In- 
variantentheorie "  in  the  Jahresb.  d.  d  Math.  Virci)iiguiig,  Vol.  I,  1890-91 
v>\j.  79-292.     See  p.  99. 


ALGEBRA  347 

linear  systems  of  bilinear  and  quadratic  forms,  L,  Kronecker  who  ex- 
tended the  researches  of  Weierstrass  and  had  a  controversy  with  C. 
Jordan  on  certain  discordant  results,  J.  G.  Darboux  who  in  1S74  gave 
a  general  and  elegant  derivation  of  theorems  due  to  K.  Weierstrass 
and  L.'  Kronecker,  G.  Frobenius  who  applied  the  transformation  of 
bilinear  forms  to  "Pfaff's  problem":  To  determine  when  two  given 
linear  differential  expressions  of  n  terms  can  be  converted  one  into 
the  other  by  subjecting  the  variables  to  general  point  transformations. 
The  study  of  invariance  of  quadratic  and  bilinear  forms  from  the 
stand-point  of  group  theory  was  pursued  by  H.  Werner  (1S89),  S. 
Lie  (1SS5)  and  W.  Killing  (1S90).  Finite  binary  groups  were  exam- 
ined by  H.  A.  Schwarz  (1S71),  and  Felbc  Klein.  Schwarz  is  led  to  the 
problem,  to  find  "all  spherical  triangles  whose  symmetric  repetitions 
on  the  surface  of  a  sphere  give  rise  to  a  finite  number  of  spherical 
triangles  diiTering  in  position,"  and  deduces  the  forms  belonging 
thereto.  W^ithout  a  knowledge  of  what  Schwarz  and  W.  R.  Hamilton 
had  done,  Klein  was  led  to  a  determination  of  the  finite  binary  linear 
groups  and  their  forms.  Representing  transformaticns  as  motions 
and  adopting  Riemann's  interpretation  of  a  complex  variable  on  a 
spherical  surface,  F.  Klein  sets  up  the  groups  of  those  rotations  which 
bring  the  five  regular  solids  into  coincidence  with  themselves,  and  the 
accompanying  forms.  The  tetrahedron,  octahedron  and  icosahedron 
lead  respectively  to  12,  24  and  60  rotations;  the  groups  in  question 
were  studied  by  Klein.  The  icosahedral  group  led  to  an  icosahedral 
equation  which  stands  in  intimate  relation  with  the  general  equation 
of  the  fifth  degree.  Klein  made  the  icosahedron  the  centre  of  his  theory 
of  the  quintic  as  given  in  his  Vorlesimgen  iiber  das  Ikosaeder  utid  die 
Auflosung  der  Gleichungen  fiinften  Grades,  Leipzig,  1884. 

Finite  substitution  groups  and  their  forms,  as  related  to  linear 
differential  equations,  were  investigated  by  R.  Fuchs  (Crelle,  66,  68) 
in  1866  and  later.  If  the  equation  has  only  algebraic  integrals,  then 
the  group  is  finite,  and  conversely.  Fuchs's  researches  on  this  topic 
were  continued  by  C.  Jordan,  F.  Klein,  and  F.  Brioschi.  Finite  ternary 
and  higher  groups  have  been  studied  in  connection  with  invariants  by 
F.  Klein  who  in  1887  made  two  such  groups  the  basis  for  the  solution 
of  general  equations  of  the  sbcth  and  seventh  degrees.  In  1886  F.  N. 
Cole,  under  the  guidance  of  Klein,  had  treated  the  sextic  equation  in 
the  Am.  Jour,  of  Math.,  Vol.  8.  The  second  group  used  by  Klein  was 
studied  with  reference  to  the  140  lines  in  space,  to  which  it  leads,  by 
H.  Maschke  in  i8go. 

The  relationship  of  invariantal  forms,  the  study  of  which  was 
initiated  by  A.  Cayley  and  J.  J.  Sylvester,  received  since  1868  em- 
phasis in  the  writings  of  R.  F.  A.  Clebsch  and  P.  Gordan.  Gordan 
proved  in  Crelle,  Vol.  69,  the  finiteness  of  the  system  for  a  single 
binary  form.  This  is  known  as  "Gordan's  theorem."  Even  in  the 
later  simplified  forms  the  proof  of  it  is  involved,  but  the  theorem 


348  A  HISTORY  OF  MATHEMATICS 

Anelds  practical  methods  to  determine  the  existing  systems.  G. 
Peano  in  1881  generalized  the  theorem  and  applied  it  to  the  "cor- 
respondences" represented  by  certain  double-binary  forms.  In  1890 
D.  Hilbert,  by  using  only  rational  processes,  demonstrated  the  finite- 
ness  of  the  system  of  invariants  arising  from  a  given  series  of  any  forms 
in  n  variables.  A  modification  of  this  proof  which  has  some  advantages 
was  given  by  W.  E.  Story  of  Clarke  University.  Hilbert's  research 
bears  on  the  number  of  relations  called  syzygies,  a  subject  treated  be- 
fore this  time  by  A.  Cayley,  C.  Hermite,  F.  Brioschi,  C.  Stephanos 
of  Athens,  J.  Hammond,  E.  Stroh,  and  P.  A.  MacMahon. 

The  symbolic  notation  in  the  theory  of  invariants,  introduced  by 
S.  H.  Aronhold  and  R.  F.  A.  Clebsch,  was  developed  further  by  P.  Gor- 
dan,  E.  Stroh,  and  E.  Study  in  Germany.  English  writers  endeavored 
to  make  the  expressions  in  the  theory  of  forms  intuitively  evident  by 
graphic  representation,  as  when  Sylvester  in  1878  uses  the  atomic 
theory,  an  idea  applied  further  by  W.  K.  Clifford.  The  symbolic 
method  in  the  theory  of  invariants  has  been  used  by  P.  A.  MacMahon 
in  the  article  "Algebra"  in  the  eleventh  edition  of  the  EncydopcBdia 
Britannica,  and  by  J.  H.  Grace  and  A.  Young  in  their  Algebra  of  In- 
variants, Cambridge,  1903.  Using  a  method  in  C.  Jordan's  great 
memoirs  on  invariants,  these  authors  are  led  to  novel  results,  notably 
to  "an  exact  formula  for  the  maximum  order  of  an  irreducible  co- 
variant  of  a  system  of  binary  forms."  A  complete  syzygetic  theory 
of  the  absolute  orthogonal  concomitants  of  binary  quantics  was  con- 
structed by  Edwin  B.  Elliott  of  Oxford  by  a  method  that  is  not  sym- 
bolic, while  P.  A.  MacMahon  in  1905  employs  a  symbolic  calculus  in- 
volving imaginary  umbrae  for  similar  purposes.  While  the  theory 
of  invariants  has  played  an  important  role  in  modern  algebra  and 
analytic  projective  geometry,  attention  has  been  directed  also  to  its 
emplo3rment  in  the  theory  of  numbers.  Along  this  line  are  the  re- 
searches of  L.  E.  Dickson  in  the  Madison  Colloquium  of  19 13. 

The  establishment  of  criteria  by  means  of  which  the  irreducibility 
of  expressions  in  a  given  domain  may  be  ascertained  has  been  inves- 
tigated by  F.  T.  V.  Schubert  (1793),  K.  F.  Gauss,  L.  Kronecker,  F.  W. 
P.  Schonemann,  F.  G.  M.  Eisenstein,  R.  Dedekind,  G.  Floquet,  L. 
Konigsberger,  E.  Netto,  O.  Perron,  M.  Bauer,  W.  Dumas  and  H. 
Blumberg.  The  theorem  of  Schonemann  and  Eisenstein  declares  that 
if  the  polynomial  x'^+ci  x"'^+  .  .  .  +c„  with  integral  coefficients  is 
such  that  a  prime  p  divides  every  coefficient  Ci,  .  .  ,  c„,  but  p^  does 
not  go  into  c„,  then  the  polynomial  is  irreducible  in  the  domain  of 
rational  numbers.  This  theorem  may  be  regarded  as  the  nucleus  of 
the  work  of  the  later  authors.  Floquet  and  Konigsberger  do  not 
limit  themselves  to  polynomials,  but  consider  also  linear  homogeneous 
differential  expressions.  Blumberg  gives  a  general  theorem  which 
practically  includes  all  earher  results  as  special  cases. ^ 

'  For  bibliography  see  Trans.  Am.  Math.  Soc,  Vol.  17,  19x6,  pp.  517-544. 


.\LGEBRA  34t* 

Theory  oj  Equations  and  Theory  if  Gtoups 

A  notable  event  was  the  reduction  of  the  quintic  equation  to  the 
trinomial  form,  effected  by  George  Birch  Jcrrard  (  ?-iS63)  in  his 
Mathematical  Researches  (1S32-1S35).  Jerrard  graduated  B.  A.  at 
Trinity  College,  Dublin,  in  1827.  It  was  not  until  1861  that  it  became 
generally  knoMTi  that  this  reduction  had  been  effected  as  early  as 
1786  by  Erland  Samuel  Bring  (1736-170'),  a  Swede,  and  brought 
out  in  a  pul:)lication  of  the  University  of  Lund.  Both  Bring  and  Jer- 
rard used  the  method  of  E.  W.  Tschimhausen.  Bring  nc\-er  claimed 
that  his  transformation  led  to  the  general  algebraic  solution  of  the 
quintic,  but  Jerrard  ]~)ersi:.tcd  in  making  su:h  a  claim  even  after 
N.  H.  Abel  and  others  had  o.Tered  proofs  establishing  the  impossibility 
of  a  general  solution.  In  1^,36,  WiUiam  R.  Hamilton  made  a  report 
on  the  validity  of  Jerrard's  method,  and  showed  that  by  his  process 
the  quintic  could  be  transformed  to  any  one  of  the  four  trinomial 
forms.  Hamilton  defined  the  limits  of  its  applicability  to  higher  equa- 
tions. J.  J.  Sylvester  investigated  this  question,  What  is  the  lowest 
degree  an  equation  can  have  in  order  that  it  may  admit  of  being 
deprived  of  i  consecutive  terms  by  aid  of  equations  not  higher  than 
nh  degree.  He  carried  the  investigation  as  far  as  7  =  8,  and  was  led 
to  a  series  of  numbers  which  he  named  "Hamilton's  numbers."  A 
transformation  of  equal  importance  to  Jerrard's  is  that  of  Sylvester, 
who  expressed  the  quintic  as  the  sum  of  three  fifth-powers.  The 
covariants  and  invariants  of  higher  equations  have  been  studied 
much  in  recent  years. 

In  the  theory  of  equations  J.  L.  Lagrange,  J.  R.  Argand,  and  K.  F. 
Gauss  furnished  proof  to  the  important  theorem  that  every  algebraic 
equation  has  a  real  or  a  complex  root.  N.  H.  Abel  proved  rigorously 
that  the  general  algebraic  equation  of  the  fifth  or  of  higher  degrees 
cannot  be  solved  by  radicals  {Crelle,  I,  1826).  Before  Abel,  an  ItaUan 
physician,  Paolo  Ruffini  (1765-1S22),  had  printed  a  proof  of  the  in- 
solvability.  It  appears  in  his  book,  Teoria  generale  delle  eqiiazioni, 
Bologna,  17QQ,  and  in  later  articles  on  this  subject.  Ruffini's  proof 
was  criticised  by  his  countryman,  G.  F.  Malfatti.  L.  N.  M.  Carnot, 
A.  M.  Legendre,  and  S.  D.  Poisson,  in  a  report  of  1813  on  a  paper  of 
A.  L.  Cauchy,  had  occasion  to  refer  to  Ruffini's  proof  as  "fondee  sur 
des  raisonnemens  trop  vagues,  et  n'ait  pas  ete  generalement  admise."  ^ 
N.  H.  Abel  remarked  thai'  Ruffini's  reasoning  did  not  always  seem 
rigorous.  But  Cauchy  in  1821  wrote  to  Ruffini  that  he  had  "demontre 
completement  I'insolubilite  algebrique  des  equations  generales  d'  un 
degre  superieur  au  quatrieme."  ^  J.  Hecker  showed  in  1886  that 
Ruffini's  proof  was  sound  in  general  outline,  but  faulty  in  some  of 
the  detail.^    E.  Bortolotti  in  1902  stated  that  Ruffini's  proof,  as  given 

'  E.  Bortolotti,  Cartcggio  di  Paolo  Ruffini,  Roma,  iqo6,  p.  32. 

*E.  Bortolotti,  Influenza  dcW  opera  mat.  di  P.  Ruffini,  1902,  p.  34. 

'J.  Hecker,  Ueber  Ruffini's  Bcweis  (Dissertation),  Bonn,  18S6. 


350  A  HISTORY  OF  MATHEMATICS 

in  1S13  in  his  book  Rcflessioni  iniorno  alia  sohizione  aeW  equazioni 
algebraichc,  was  substantially  the  same  as  that  given  later  by  Pierre 
Laurent  Wantzcl  ^  (1S14-1S4S),  but  only  the  second  part  of  Wantzel's 
simplified  proof  resembles  Ruffini's;  the  first  part  is  modelled  after 
Abel's.  Wantzel,  by  the  way,  deserves  credit  for  having  given  the 
first  rigorous  proofs  {Liouville,  Vol.  2,  1837,  p.  366.)  of  the  im- 
possibility of  the  trisection  of  any  given  angle  by  means  of  ruler  and 
compasses,  and  of  avoiding  the  "irreducible  case"  in  the  algebraic 
solution  of  irreducible  cubic  equations.  Wantzel  was  repetiteur  at 
the  Polytechnic  School  in  Paris.  As  a  student  he  excelled  both  in 
mathematics  and  languages.  Saint-Venant  said  of  him:  "Ordinarily 
he  worked  evenings,  not  lying  down  until  late;  then  he  read,  and  took 
only  a  few  hours  of  troubled  sleep,  making  alternately  wrong  use  of 
coffee  and  opium,  and  taking  his  meals  at  irregular  hours  until  he 
was  married.  He  put  unlimited  trust  in  his  constitution,  very  strong 
by  nature,  which  he  taunted  at  pleasure  by  all  sorts  of  abuse.  He 
brought  sadness  to  those  who  mourn  his  premature  death," 

Ruffini's  researches  on  equations  are  remarkable  as  containing 
anticipations  of  the  algebraic  theory  of  groups.^  Ruffini's  "per- 
mutation" corresponds  to  our  term  "group."  He  di\'ided  groups  into 
"simple"  and  "complex,"  and  the  latter  into  intransitive,  transitive 
imprimitive,  and  transitive  primitive  groups.  He  established  the 
important  theorem  for  which  the  name  "Ruffini's  theorem"  has 
been  suggested,^  that  a  group  does  not  necessarily  have  a  subgroup 
whose  order  is  an  arbitrary  divisor  of  the  order  of  the  group.  The 
collected  works  of  Ruffini  afe  published  under  the  auspices  of  the 
Circolo  Matematico  di  Palermo;  the  first  volume  appeared  in  1915 
with  notes  by  Ettore  Bortolotti  of  Bologna.  A  transcendental  solu- 
tion of  the  quintic  involving  elliptic  integrals  was  given  by  Ch.  Her- 
mite  {Compt.  Rotd.,  185S,  1S65,  1866).  After  Hermite's  first  publica- 
tion, L.  Kronecker,  in  1858,  in  a  letter  to  Hermite,  gave  a  second 
solution  in  which  was  obtained  a  simple  resolvent  of  the  sixth  degree. 

Abel's  proof  that  higher  equations  cannot  always  be  solved  alge- 
braically led  to  the  inquiry  as  to  what  equations  of  a  given  degree 
can  be  solved  by  radicals.  Such  equations  are  the  ones  discussed  by 
K.  F.  Gauss  in  considering  the  division  of  the  circle.  Abel  advanced 
one  step  further  by  proving  that  an  irreducible  equation  can  always 
be  solved  in  radicals,  if,  of  two  of  its  roots,  the  one  can  be  expressed 
rationally  in  terms  of  the  other,  provided  that  the  degree  of  the  equa- 
tion is  prime;  if  it  'v  not  prime,  then  the  solution  depends  upon  that 
of  equations  of  lower  degree.     Through  geometrical  considerations, 

1  E.  Bortolotti,  Influenza,  etc.,  1902,  p.  26.  Wantzel's  proof  is  given  in  Noiivelles 
Annates  Mathemaliques,  Vol.  4,  1845,  pp.  57-65.  See  also  Vol.  2,  pp.  1 17-127.  The 
second  part  of  Wantzel's  proof,  involv'ng  substitution-theory,  is  reproduced  in  J.  A. 
Serret's  Algebre  superienre. 

2H.  Burkhardt,  in  Zeilschr.f.  MathemaiiJi  u.  Physik,  Suppl.,  1892. 

3  G.  A.  Miller,  in  Bibtiothcca  mallicmatica,  3.  F.  Vol.  10,  1909-1910,  p.  318. 


alc;kbra  351 

L.  O.  Hesse  came  upon  algebraically  solvable  equations  of  the  ninth 
cL'gree,  not  included  in  the  pre\-ious  grou])s.  The  subject  was  power- 
fully advanced  in  Paris  by  the  youthful  Evaristc  Galois  (1811-1832).^ 
He  was  born  at  Bourg-la-Reine,  near  Paris.  He  began  to  exhibit 
most  extraordinary  mathematical  genius  after  his  iifteenth  year. 
His  was  a  short ,^  sad  and  pestered  life.  He  was  twice  refused  ad- 
mittance to  the  Ecole  Polytechnique,  on  account  of  inability  to  meet 
the  (to  him)  trivial  demands  of  examiners  who  failed  to  recognize 
his  genius.  He  entered  the  Ecole  Normale  in  1829,  then  an  inferior 
school.  Proud  and  arrogant,  and  unable  to  see  the  need  of  the  cus- 
tomary detailed  explanations,  his  career  in  that  school  was  not  smooth. 
Drawn  into  the  turmoil  of  the  revolution  of  1830,  he  was  forced  to 
leave  the  Ecole  Normale.  After  several  months  spent  in  prison,  he 
was  killed  in  a  duel  over  a  love  affair.  Ordinary  text-books  he  dis- 
posed of  as  rapidly  as  one  would  a  novel.  He  read  J.  L.  Lagrange's 
memoirs  on  equations,  also  writings  of  A.  M.  Legendre,  C.  G.  J. 
Jacobi,  and  N.  H.  Abel.  As  early  as  the  seventeenth  year  he  reached 
r:-sults  of  the  highest  importance.  Two  memoirs  presented  to  the 
A::ademy  of  Sciences  were  lost.  A  brief  paper  on  equations  in  the 
BuUdin  de  Fcrussac,  1S30,  Vol.  XIH,  p.  428,  gives  results  which  seem 
to  be  applications  of  a  general  theory.  The  night  before  the  duel  he 
wrote  his  scientific  testament  in  the  form  of  a  letter  to  Auguste  Cheva- 
li;.'r,  containing  a  statement  of  the  mathematical  results  he  had  reached 
and  asking  that  the  letter  be  published,  that  "Jacobi  or  Gauss  pass 
judgment,  not  on  their  correctness,  but  on  their  importance."  Two 
memoirs  found  among  his  papers  were  published  by  J.  Liouville  in 
1846.  Further  manuscripts  Avere  published  by  J.  Tannery  at  Paris 
in  1908.  As  a  rule  Galois  did  not  fully  prove  his  theorems.  It  was 
only  with  difi&culty  that  Liouville  was  able  to  penetrate  into  Galois' 
idjas.  Several  commentators  worked  on  the  task  of  filling  out  the 
lacuna;  in  Galois'  exposition.  Galois  was  the  first  to  use  the  word 
"group"  in  a  technical  sense,  in  1830.  He  divided  groups  into  simple 
and  compound,  and  observed  that  there  is  no  simple  group  of  any 
composite  order  less  than  60.  The  word  "group"  was  used  by  A. 
Cayley  in  1S54,  by  T.  F.  Kirkman  and  J.  J.  Sylvester  in  1860.^  Galois 
proved  the  important  theorem  that  every  invariant  subgroup  gives 
rise  to  a  quotient  group  which  exhibits  many  fundamental  properties 
of  the  group.  He  showed  that  t^^  each  algebraic  equation  corresponds 
a  group  of  substitutions  which  reflects  the  essential  character  of  the 
equation.  In  a  paper  published  in  1S46  he  established  the  beautiful 
theorem:  In  order  thai"  an  irreducible  equation  of  prime  degree  be 
solvable  by  radicals,  it  is  necessary  and  sufiicient  that  all  its  roots  be 

'  See  life  by  Paul  Dupuy  m  Annates  de  Vicole  normale  supirietire,  3.  S.,  Vol.  XTTT, 
1896.  See  also  E.  Picard,  Ociivrcs  math,  d'  J^varisle  Galois,  Paris,  1S97;  J.  Pierpor.t 
Bulletin  Am.  Math.  Soc,  2.  S.,  Vol.  IV,  189S,  pp.  332-340. 

*  G.  A.  Miller  in  Am.  Math.  Monthly,  Vol.  XX,  1913,  p.  18. 


352  A  HISTORY  OF  MA1"Hf:MAllCS 

rational  in  any  two  of  them.  Galois'  use  of  substitution  groups  to 
determine  the  algebraic  solvability  of  equations,  and  N.  H.  Abel's 
somewhat  earlier  use  of  these  groups  to  prove  that  general  equations 
of  degrees  higher  than  the  fourth  cannot  be  solved  by  radicals,  fur- 
nished strong  incentives  to  the  vigorous  cultivation  of  group  theory. 
It  was  A.  L.  Cauchy  who  entered  this  field  next.  To  Galois  are  due 
also  some  valuable  results  in  relation  to  another  set  of  equations, 
presenting  themselves  in  the  theory  of  elliptic  functions,  viz.,  the 
modular  equations.  To  Cauchy  has  been  given  the  credit  of  being 
the  founder  of  the  theory  of  groups  of  finite  order, ^  even  though  funda- 
mental results  had  been  previously  reached  by  J.  L.  Lagrange,  Pietro 
Abbati  (i 786-1842),  P.  Ruffini,  N.  H.  Abel,  and  Galois.  Cauchy's 
first  publication  was  in  181 5,  when  he  proved  the  theorem  that  the 
number  of  distinct  values  of  a  non-symmetric  function  of  degree  n 
cannot  be  less  than  the  largest  prime  that  divides  n,  without  becom- 
ing equal  to  2.  Cauchy's  great  researches  on  groups  appeared  in  his 
Exercises  d' analyse  et  de  physique  matheniatique,  1S44,  and  in  articles 
in  the  Paris  Comptes  Rendus,  1845-1846.  He  did  not  use  the  term 
"group,"  but  he  uses  (x  y  z  u  v  w)  and  other  devices  to  denote  sub- 
stitutions, uses  the  terms  "cyclic  substitution,"  "order  of  a  substitu- 
tion," "identical  substitution,"  "transposition,"  "transitive,"  "in- 
transitive." In  1844  he  proved  the  fundamental  theorem  (stated 
but  not  proved  by  E.  Galois)  which  is  known  as  "  Cauchy's  theorem": 
Every  group  whose  order  is  divisible  by  a  given  prime  number  p  must 
contain  at  least  one  subgroup  of  order  p.  This  theorem  was  later 
extended  by  L.  Sylow.  A.  L.  Cauchy  was  the  first  to  enumerate  the 
orders  of  the  possible  groups  whose  degrees  do  not  exceed  six,  but  this 
enumeration  was  incomplete.  At  times  he  fixed  attention  on  prop- 
erties of  groups  without  immediate  concern  as  regards  applications, 
and  thereby  took  the  first  steps  toward  the  consideration  of  abstract 
groups.  In  1846  J.  Liouville  made  E.  Galois'  researches  better  known 
by  publication  of  two  manuscripts.  At  least  as  early  as  1848  J.  A. 
Serret  taught  group  theory  in  Paris.  In  1852,  Enrico  Betti  of  the 
University  of  Pisa  published  in  the  Annali  of  B.  Tortolini  the  first 
rigorous  exposition  of  Galois'  theory  of  equations  that  made  the 
theory  intelligible  to  the  general  public.  The  first  account  of  it  given 
in  a  text-book  on  algebra  is  in  the  third  edition  of  J.  A.  Serret's  Alge- 
bre,  1866. 

In  England  the  earliest  studies  in  group  theory  are  due  to  Arthur 
Cayley  and  William  R.  Hamilton.  In  1854  A.  Cayley  published  a 
paper  in  the  Philosophical  Magazine  which  is  usually  accepted  as 
founding  the  theory  of  abstract  groups,  although  the  idea  of  abstract 
groups  occurs  earlier  in  the  papers  of  A.  L.  Cauchy,  and  Cayley's 

1  Our  account  of  Cauchy's  researches  on  groups  is  drawn  from  the  article  of  G.  A. 
Miller  in  Bihliothcca  Ma'thcmatica,  Vol.  X,  1909-1010,  pp.  317-329,  and  that  of 
Tosephine  E.  Burns  in  Am.  Math.  Montldy,  Vol.  XX,  191.^,  pp.  141-148. 


ALGEBRA  353 

article  is  not  entirely  abstract.  Formal  definitions  of  abstract  groups 
were  not  given  until  later,  by  L.  Kronecker  (1870),  H.  Weber  (1882), 
and  G.  Frobenius  (1887).  The  transition  from  substitution  groups 
to  abstract  groups  was  gradual.^  It  may  be  recalled  here  that,  before 
1854,  there  were  two  sources  from  which  the  theory  of  groups  of  finite 
order  originated.  In  the  writings  of  J.  Lagrange,  P.  Ruffini,  N.  H. 
Abel,  and  E.  Galois  it  sprang  from  the  theory  of  algebraic  equations. 
A  second  source  is  the  theory  of  numbers;  the  group  concept  is  funda- 
mental in  some  of  L.  Euler's  work  on  power  residuts  and  in  some  of 
the  early  work  of  K.  F.  Gauss.  It  has  been  pointed  out  more  recently 
that  the  group  idea  really  underlies  geometric  transformations  and 
is  implied  in  Euclid's  demonstrations.-  Abstract  groups  are  con- 
sidered apart  from  any  of  their  applications. 

A.  Cayley  illustrates  his  paper  of  1854  by  means  of  the  laws  of 
combination  of  quaternion  imaginaries,  quaternions  having  been 
invented  by  William  R.  Hamilton  eleven  years  previously.  In  1859 
Cayley  pointed  out  that  the  quaternion  units  constitute  a  group  of 
order  8,  now  known  as  the  quaternion  group,  when  they  are  multiplied 
together.^  William  R.  Hamilton,  without  using  the  technical  lan- 
guage of  group  theory,  developed  in  1856,  in  his  study  of  a  new  system 
of  roots  of  unity,  the  properties  of  the  groups  of  the  regular  solids, 
as  generated  by  two  operators  or  elements,  and  he  proved  that  these 
groups  may  be  completely  defined  by  the  orders  of  their  two  generating 
operators  and  the  order  of  their  product. 

E.  Picard  puts  the  matter  thus:  "A  regular  polyhedron,  say  an 
icosahedron,  is  on  the  one  hand  the  solid  that  all  the  world  knows;  it 
is  also,  for  the  analyst,  a  group  of  finite  order,  corresponding  to  the 
divers  ways  of  making  the  polyhedron  coincide  with  itself.  The  in- 
vestigation of  all  the  types  of  groups  of  motion  of  finite  order  interests 
not  alone  the  geometers,  but  also  the  crystallographers;  it  goes  back 
essentially  to  the  study  of  groups  of  ternary  linear  substitutions  of 
determinant  +1,  and  leads  to  the  thirty-two  systems  of  symmetry 
of  the  crystallographers  for  the  particular  complex." 

In  1858  the  Institute  of  France  offered  a  prize  for  a  research  on 
group  theory  v/hich,  though  not  awarded,  stimulated  research.  In 
1859  Emile  Leonard  Mathieu  (1835-1890)  of  the  University  of  Nancy 
wrote  a  thesis  on  substitution  groups,  while  in  i860  Camille  Jordan 
(1838-  J  of  the  Ecole  Poly  technique  in  Paris  contributed  the  first 
of  a  series  of  papers  which  culminated  in  his  great  Traite  des  substitu- 
tions, 1870.  Jordan  received  his  doctorate  in  1861  in  Paris;  he  is  editor 
of  the  Journal  de  mathcmatiques  purcs  ct  appliquces.  His  first  paper 
on  groups  gives  the  fundamental  theorem  that  the  total  number  of 

>  G.  A.  Miller,  in  Bibliolhcca  Mathcmatica,  3.  Ed.,  Vol.  XX,  1909-1910,  p.  326. 
We  are  making  much  use  of  Miller's  historical  sketch. 
*  H.  Poincar^  in  Monist,  Vol.  9,  1898,  p.  34. 
'G.  A.  Miller  in  Bibliolheca  Mathemalica,  Vol.  XI,  1910-1911,  pp.  314-311;. 


354  A  HISTORY  OF  MATHEMATICS 

substitutions  of  n  letters  which  are  commutative  with  every  substitu- 
tion of  a  regular  group  G  on  the  same  n  letters  constitute  a  group 
which  is  similar  to  G.  To  Jordan  is  due  the  fundamental  concept  of 
class  of  a  substitution  group  and  he  proved  the  constancy  of  the 
factors  of  composition.  He  also  proved  that  there  is  a  finite  number 
of  primitive  groups  whose  class  is  a  given  number  greater  than  3; 
and  that  the  necessary  and  sufficient  condition  that  a  group  be  solvable 
is  that  its  factors  of  composition  are  prime  numbers.^  Prominent 
among  C.  Jordan's  pupils  is  Edmont  Maillei  (1865-  ),  editor  of 
Llnterm'diaire  des  malhematiciens ,  who  has  made  extensive  contribu- 
tions. 

In  Germany  L.  Kronecker  and  R.  Dedekind  were  the  earhest  to 
become  acquainted  with  the  Galois  theory.  Kronecker  refers  to  it 
in  an  article  published  in  1853  in  the  Berichte  of  the  Berlin  Academy. 
Dedekind  lectured  on  it  in  Gottingen  in  1858.  In  1879-1880  E.  Netto 
gave  lectures  in  Strassburg.  His  Substitutionstheorie,  1882,  was  trans- 
lated into  Italian  in  1885  by  Giuseppe  Battaglini  (1826-1894)  of  the 
University  of  Rome,  and  into  English  in  1892  by  F.  N.  Cole,  then  at 
Ann  Arbor.  The  book  placed  the  subject  within  easier  reach  of  the 
mathematical  public. 

In  1862-1863  Ludwig  Sylow  (1832-1Q18)  gave  lectures  on  substitu- 
tion groups  in  Christiania,  Norway,  which  were  attended  by  Sophus 
Lie.  Extending  a  theorem  given  nearly  thirty  years  earlier  by  A.  L. 
Cauchy,  Sylow  obtained  the  theorem  known  as  "Sylow's  theorem": 
Every  group  whose  order  is  divisible  by  p^"-,  but  not  by  p'^+^j  p  being  a 
prime  number,  contains  i+kp  subgroups  of  order  p'^.  About  twenty 
years  later  this  theorem  was  extended  still  further  by  Georg  Frobenius 
(1849-1917)  of  the  University  of  Berlin,  to  the  effect  that  the  number 
of  subgroups  is  kp+i,  k  being  an  integer,  even  when  the  order  of  the 
group  is  divisible  by  a  higher  power  of  p  than  /)™.  Sophus  Lie  took 
a  very  important  step  by  the  explicit  application  of  the  group  concept 
to  new  domains  and  the  creation  of  the  theory  of  continuous  groups. 
Marius  Sophus  Lie  (1842-1899)  -  was  born  in  Nordfjordeide  in  Nor- 
way. In  1859  he  entered  the  University  of  Christiania,  but  not  until 
1868  did  this  slowly  developing  youth  display  marked  interest  in 
mathematics.  The  waitings  of  J.  V.  Poncelet  and  J.  Pliicker  awakened 
his  genius.  In  the  winter  of  1869-1S70  he  met  Felix  Klein  in  Berhn 
and  they  published  some  papers  of  joint  authorship..  The  summer  of 
1870  they  were  together  in  Paris  where  they  were  in  close  touch  with 
C.  Jordan  and  J.  G.  Darboux.  It  was  then  that  Lie  discovered  his 
contact-transformation  which  changes  the  straight  lines  of  ordinary 
space  over  into  spheres.  This  led  him  to  a  general  theory  of  trans- 
formation.    At  the  outbreak  of  the  Franco-Prussian  war,  F.  Klein 

'  (J.  A.  Miller  in  Bihliotheca  maihematica,  t,.  S.,  Vol.  X,  1909-1Q10,  p.  .^23. 
2  F.  EiiRel  in  Biblinthcca  malhcmalica,  3.  S.,  Vol.  I,  1900,  pp.  160-204;  M.  N5ther 
in  Matk.  Annalen,  Vol.  53,  pp.    1-41. 


ALGEBRA  355 

left  Paris;  S.  Lie  started  to  travel  afoot  throupjli  France  into  Italy, 
but  was  arrested  as  a  spy  and  imprisoned  for  a  month  until  Darboux 
was  able  to  secure  his  release.  In  1872  he  was  elected  professor  at  the 
University  of  Christiania,  with  all  his  time  available  for  research.  In 
1S71-1872  he  entered  upon  the  study  of  partial  differential  equations 
of  the  first  order,  and  in  1873  he  arrived  at  the  theory  of  transforma- 
tion groups,  according  to  which  finite  continuous  groups  are  applied 
to  infinitesimal  transformations.  He  considered  a  very  general  and 
important  kind  of  transformations  called  contact-transformations, 
and  their  application  in  the  theory  of  partial  differential  equations 
of  the  first  and  second  orders.  As  his  group  theory  and  theories  of 
integration  met  with  no  appreciation,  he  returned  in  1876  to  the  study 
of  geometry — minimal  surfaces,  the  classification  of  surfaces  according 
to  the  transformation  group  of  their  geodetic  lines.  The  starting  of 
a  new  journal,  the  Archiv  for  Mathematik  og  Naturvidenskab,  in  1876, 
enabled  him  to  publish  his  results  promptly.  G.  H.  Halphen's  pub- 
lications of  1882  on  difTerential  invariants  induced  Lie  to  direct  at- 
tention to  his  own  earlier  researches  and  their  greater  generaUty.  In 
1884  Friedrich  Engel  was  induced  by  F.  Klein  and  A.  Mayer  to  go  to 
Christiania  to  assist  Lie  in  the  preparation  of  a  treatise,  the  Theorie 
der  Transformationsgriippen,  1888-1893.  Lie  accepted  in  18S6  a 
professorship  at  the  University  of  Leipzig.  In  1889-1890  over-work 
led  to  insomnia  and  depression  of  spirits.  While  he  soon  recovered 
his  power  for  work,  he  ever  afterwards  was  over-sensitive  and  mis- 
trustful of  his  best  friends.  With  the  aid  of  Engel  he  published  in 
i8gi  a  memoir  on  the  theory  of  infinite  continuous  transformation 
groups.  In  1898  he  returned  to  Norway  where  he  died  the  following 
year.  Lie's  lectures  on  Dijferentialgleichungen,  given  in  Leipzig,  were 
brought  out  in  book  form  by  his  pupil,  Georg  Schefers,  in  1891.  In 
1895  F.  Klein  declared  that  Lie  and  H.  Poincare  were  the  two  most 
active  mathematical  investigators  of  the  day.  The  following  quota- 
tion from  an  article  written  by  Lie  in  1895  indicates  how  his  whole 
soul  was  permeated  by  the  group  concept:^  "In  this  century  the 
concepts  known  as  substitution  and  substitution  group,  transforma- 
tion and  transformation  group,  operation  and  operation  group, 
invariant,  differential  invariant,  and  differential  parameter,  appear 
continually  more  clearly  as  the  most  important  concepts  of  mathe- 
matics. While  the  curve  as  the  representation  of  a  function  of  a 
single  variable  has  been  the  most  important  object  of  mathematical 
investigation  for  nearly  two  centuries  from  Descartes,  while  on  the 
other  hand,  the  concept  of  transformation  first  appeared  in  this 
century  as  an  expedient  in  the  study  of  curves  and  surfaces,  there 
has  gradually  developed  in  the  last  decades  a  general  theory  of  trans- 
formations whose  elements  are  represented  by  the  transformation 

'  Berichlcd.  Kornigl.  Saechs.  Geselhrhuft,  1895;  translated  Ijy  G.  A.  Miller  in  Am. 
Math.  Monthly,  Vol.  Ill,  1896,  p.  296. 


356  A  HISTORY  OF  MATHEMATICS 

itself  while  the  series  of  transformations,  in  particular  the  transforma- 
tion groups,  constitute  the  object." 

In  close  association  with  S.  Lie  in  the  advancement  of  group  theory 
and  its  appHcations  was  Felix  Klein  (1849-  )•  He  was  born  at 
DUsseldorf  in  Prussia  and  secured  his  doctorate  at  Bonn  in  1868.  After 
studying  in  Paris,  he  became  privat-docent  at  Gottingen  in  1871, 
professor  at  Erlangen  in  1S72,  at  the  Technical  High  School  in  Mu- 
nich in  1875,  9-t  Leipzig  in  iSSo  and  at  Gottingen  in  1886.  He  has  been 
active  not  only  in  the  advancement  of  various  branches  of  mathemat- 
ics, but  also  in  work  of  organization.  Famous  for  laying  out  lines  of 
research  is  his  Erlangen  paper  of  1872,  Vergleichende  Betrachtungen 
iiber  neuere  geometrische  Forschungen.  He  became  member  of  the  com- 
mission on  the  publication  of  the  Encyklopadie  der  mathetnatischen 
Wissenschaftcn  and  editor  of  the  fourth  volume  on  mechanics,  also 
editor  of  Mathematischc  Aniialcn,  1877,  and  in  1908  president  of  the 
International  CommJssicn  on  the  Teaching  of  Mathematics.  As  an 
inspiring  lecturer  on  mathematics  he  has  wielded  a  wide  influence 
upon  German  and  American  students.  About  191 2  he  was  forced  by 
ill-health  to  discontinue  his  lectures  at  Gottingen,  but  in  1914  the  ex- 
citement of  the  war  roused  him  to  activity,  much  as  J.  Lagrange  was 
aroused  at  the  outbreak  of  the  French  Revolution,  and  Klein  resumed 
lecturing.  He  has  constantly  emphasized  the  importance  of  both 
schools  of  mathematical  thought,  namely,  the  intuitional  school,  and 
the  school  that  rests  everything  on  abstract  logic.  In  his  opinion, 
"the  intuitive  grasp  and  the  logical  treatment  should  not  exclude, 
but  should  sup]ilement  each  other." 

S.  Lie's  method  of  treating  differential  invariants  was  further  in- 
vestigated by  K.  Zorawski  in  Acta  Math.,  Vol.  XVI,  1892-1893.  In 
1902  C.  N.  Haskins  determined  the  number  of  functionally  independ- 
ent invariants  of  any  order,  while  A.  R.  Forsyth  obtained  the  invariants 
for  ordinary  Euclidean  space.  Differential  parameters  have  been 
investigated  by  J.  Edmund  Wright  of  Bryn  Mawr  College.^  Lie's 
theory  of  invaru^xits  of  finite  continuous  groups  was  attacked  on  logi- 
cal grounds  by  E.  Study  of  Bonn,  in  1908.  The  validity  of  this  criti- 
cism was  partly  admitted  by  F.  Engel. 

Another  method  of  treating  differential  invariants,  originally  due 
to  E.  B.  Christoffel,  has  been  called  by  G.  Ricci  and  T.  Levi-Civita 
of  Padua  "covariant  derivation,"  {M athematische  Annalen,  Vol.  54, 
1901).  A  third  method  was  introduced  by  H.  Maschke  ^  who  used  a 
symbolism  similar  to  that  for  algebraic  invariants. 

Henry  W.  Stager  published  in  19 16  A  Sylow  Factor  Table  for  the  first 
Twelve  Thousand  Numbers:  For  every  number  up  to  1200  the  divisors 
of  the  form  p(kp+i)  are  given,  where  pis  a.  prime  greater  than  2  and 

^  We  are  using  J.  E.  Wright's  Invariants  of  Quadratic  Dijferential  Forms,  1908, 
pp.  5-8. 

*  Trans.  Am.  Math.  Soc,  Vol.  i,  1900,  pp.  197-204. 


ALGEBRA 


35; 


^  is  a  positive  integer.     These  divisors  aid  in  the  determination  of 
the  number  of  Sylow  subgroups. 

Solvable  (H.  Weber's  "metacycHc")  groups  have  been  studied  by 
G.  Frobenius  who  proved  that  every  group  of  composite  order  that 
is  not  divisible  by  the  square  of  a  prime  number  must  be  compound, 
and  that  all  these  groups  are  solvable,  for  the  orders  of  their  self- 
conjugate  subgroups  and  of  their  quotient  groups  cannot  be  divisible 
by  the  square  of  a  j^rime  number.^  The  study  of  solvable  groups  has 
been  pursued  also  by  L.  Sylow,  W.  Burnside,  R.  Dedekind  (who  inves- 
tigated what  he  called  the  Hamiltonian  group),  and  G.  A.  Miller  who 
with  Frobenius  develoi)ed  about  1S93-1S96  elegant  methods  for  prov- 
ing the  solvability  of  a  given  group.  In  1S95  O-  Holder  enumerated  all 
the  insolvable  groups  whose  order  does  not  exceed  479.  In  1898  G.  A. 
Miller  gave  the  numbers  of  all  primitive  solvable  groups  whose  degree 
is  less  than  25,  also  the  number  of  insolvable  groups  which  may  be 
represented  as  substitution  groups  whose  degree  is  less  than  12.  G.  A. 
Miller  (1899)  and  Umberto  Scarpio  (1901)  of  Verona  considered 
properties  of  commutators  and  commutator  subgroups,  and  proved 
that  the  question  of  solvability  can  be  decided  by  means  of  commuta- 
tor subgrou})s.^  Commutator  groups  have  been  studied  also  by  W.  B. 
Fite  and  Ernst  Wendt.  The  characteristics  of  non-abelian  groups  were 
investigated  since  1S96  by  G.  Frobenius  in  Berlin,  the  characteristics 
of  abelian  groups  having  been  already  employed  by  j .  Lagrange  and 
P.  Dirichiet.  Characteristics  of  solvable  groups  were  studied  in 
1901  by  Frobenius.  An  enumeration  of  abstract  groups  was  made  in 
1901  by  R.  P.  Le  Vavasseur  of  Toulouse.  The  Ust  of  intransitive 
substitution  groups  of  degree  eleven  was  shown  by  G.  A.  Miller  and 
G.  H.  Ling  in  1901  to  include  1492  distinct  substitution  groups, 
wliich  is  about  500  more  than  the  number  of  degree  ten.  H.  L.  Rietz 
proved  that  a  primitive  group  of  degree  ;/  and  order  g  contains  more 
than  g/x  +  i  substitutions  of  degree  less  than  n,  x  being  the  number  of 
transitive  constituents  in  a  maximal  subgroup  of  degree  n~i.  This 
result  is  closely  related  to  investigations  of  C.  Jordan,  A.  Bochert. 
and  E.  Maillet  on  the  class  of  a  primitive  group. ^ 

The  definition  of  a  group  w^as  simplified  in  1902  by  E.  V.  Huntington 
of  Har\'ard  University.  He  pointed  out  that  the  usual  definition, 
as  given  for  instance  in  H.  Weber's  Algebra,  contains  several  redun- 
dancies, that  only  three  postulates  (four  for  finite  groups)  are  neces- 
sary, the  independence  of  which  he  established.^  Later  discussions  of 
definitions  are  due  to  Huntington  and  E.  H.  Moore. 

'  See  G.  A.  Miller's  "Report  of  Recent  Prosjress  in  the  Theory  of  Groups  of  a 
Finite  Order"  in  Btdl.  Am.  Math.  Soc,  Vol.  5,  1899  pp.  227-249,  whicn  we  are 
using. 

*  G.  A.  Miller,  "Second  Report  on  Recent  Progress  in  the  Thcorj  of  Groups  oi 
Finite  Order"  in  Bull.  Am.  Math.  Soc,  Vol.  9,  1902,  p.  108. 

^  Loc.  cit.,  p.  118. 
Bull.  Am   Math.  Soc,  Vol.  .S,  1902,  p.  296. 


]S'^  A  IIISl'ORY  OF  MATHEMATICS 

L.  E.  Dickson  ^  said  in  1900:  "When  a  problem  has  been  exhibited 
in  group  i^Iiraseology,  the  possibility  of  a  solution  of  a  certain  char- 
acter or  the  exact  nature  of  its  inherent  difficulties  is  determined  by  a 
study  of  the  group  of  the  problem.  ...  As  the  chemist  analyzes 
a  com]X)und  to  determine  the  ultimate  elements  composing  it,  so  the 
group-theorist  decomposes  the  group  of  a  given  problem  into  a  chain 
of  simple  groups.  .  .  .  Much  labor  has  been  expended  in  the  de- 
termination of  simple  groups.  For  continuous  groups  of  a  finite 
number  of  parameters,  the  problem  has  been  completely  solved  by 
W.  Killing  and  E.  J.  Cartan  (1S94),  with  the  result  that  all  such  simple 
groups,  aside  from  five  isolated  ones,  belong  to  the  systems  investi- 
gated by  Sophus  Lie,  viz.,  the  general  projective  group,  the  pro- 
jective group  of  a  linear  complex,  and  the  projective  group  leaving 
invariant  a  non-degenerate  surface  of  the  second  order.  The  cor- 
responding problem  for  infinite  continuous  groups  remains  to  be 
solved.  With  regard  to  finite  simple  groups,  the  problem  has  been 
attacked  in  two  directions.  0.  Holder,^  F.  N.  Cole,^  W.  Burnside,^ 
G.  H.  Ling,  and  G.  A.  Miller  have  shown  that  the  only  simple  groups 
of  composite  orders  less  than  2000  are  the  previously  known  simple 
grou]is  of  orders  00,  los,  300,  504,  660,  1092.  On  the  other  hand, 
various  infinite  systems  of  finite  simple  groups  have  been  determined. 
The  cyclic  groups  of  i>rime  orders  and  the  alternating  group  of  n 
letters  («>4)  have  long  been  recognized  as  simple  groups.  The  other 
known  systems  of  finite  simple  groups  have  been  discovered  in  the 
study  of  linear  groups.  Four  systems  were  found  by  C.  Jordan, 
{Traite  des  substitutions)  in  his  study  of  the  general  linear,  the  abelian, 
and  the  two  hypoabelian  groups,  the  field  of  reference  being  the  set 
of  residues  of  integers  with  respect  to  a  prime  modulus  p.  Generaliza- 
tions may  be  made  by  employing  the  Galois  field  of  order  p"'  (desig- 
nated GF  [/>"]),  composed  of  the  p"-  Galois  complexes  formed  with  a 
root  of  a  congruence  of  degree  n  irreducible  modulo  p.  Groups  of 
Hnear  substitutions  in  a  Galois  field  were  studied  by  E.  Betti,  E. 
Mathieu,  and  C.  Jordan;  but  the  structure  of  such  groups  has  been 
determined  only  in  the  past  decade.  The  simplicity  of  the  group  of 
unar}'  linear  fractional  substitutions  in  a  Galois  field  was  first  proved 
by  E.  H.  Moore  {Bulletin  Am.  Math.  Soc,  Dec,  1893)  and  shortly 
afterward  by  W.  Burnside.  The  complete  generalization  of  C.  Jor- 
dan's four  systems  of  simple  groups  and  the  determination  of  three 
new  triply-infinite  systems  have  been  made  by  the  writer"  (/.  e.  by 

1  See  L.  E.  Dickson  in  Compte  vendu  du  II.  Congr.  inkrn.,  Paris,  igoo.  Paris, 
1902,  pp.  225,  226. 

-  O.  Holder  proved  in  Math.  Annalen,  iSg2,  that  there  are  only  two  simple  groups 
of  composite  order  less  than  200.  viz.,  those  of  order  60  and  168. 

^  F.  N.  Cole  in  Am.  Jour.  Math.,  1893  found  that  theie  could  be  only  three  such 
groups  between  orders  200  and  661,  viz.,  of  orders  360,  504,  660. 

^  W.  Burnside  showed  that  there  was  only  one  simple  group  of  composite  order 
jetween  661  and  1092. 


algp:bra  359 

L.  E.  Dickson  in  1S96).  Aside  from  the  cyclic  and  alternating  groups, 
the  known  s}'stems  of  finite  simple  groups  have  been  derived  as  quo- 
tient-groups in  the  series  of  composition  of  certain  hnear  groups. 
Miss  I.  M.  Schottenfels  of  Chicago  showed  that  it  is  possible  to  con- 
struct two  simple  grouj)s  of  the  same  order. 

The  determination  of  the  smallest  degree  (the  "class")  of  any  of  the 
non-identical  substitutions  of  primitive  groups  which  do  not  include 
the  alternating  group  was  taken  up  by  C.  Jordan  and  has  been  called 
"Jordan's  problem."  It  was  continued  by  Alfred  Bochert  of  Breslau 
and  £.  Maillet,  Bochert  proved  in  1S92:  If  a  substitution  group  of 
degree  n  does  not  include  the  alternating  group  and  is  more  than 
simply  transitiv^e,  its  class  exceeds  \n  —  i,  if  it  is  more  than  doubly 
transitive  its  class  exceeds  \n  —  i,  and  if  it  is  more  than  triply  transi- 
tive its  class  is  not  less  than  ^n—i.  E.  Maillet  showed  that  when 
the  degree  of  a  primitive  group  is  less  than  202  its  class  cannot  be  ob- 
tained by  diminishing  the  degree  by  unity  unless  the  degree  is  a  power 
of  a  prime  number.  In  1900  W.  Burnside  proved  that  every  transitive 
permutation  group  in  p  symbols,  p  being  prime,  is  either  solvable 
or  doubly  transitive. 

As  regards  linear  groups,  G.  A.  Miller  wrote  in  1899  as  follows: 
"The  linear  groups  are  of  extreme  importance  on  account  of  their 
numerous  direct  applications.  Every  group  of  a  finite  order  can 
clearly  be  represented  in  many  ways  as  a  linear  substitution  group 
since  the  ordinary  substitution  (permutation)  groups  are  merely 
very  special  cases  of  the  linear  groups.  The  general  question  of  rep- 
resenting such  a  group  with  the  least  number  of  variables  seems  to 
be  far  from  a  complete  solution.  It  is  closely  related  to  that  of  de- 
termining all  the  linear  groups  of  a  finite  order  that  can  be  represented 
with  a  small  number  of  variables.  Klein  was  the  first  to  determine  all 
the  finite  binary  groups  (in  1S75)  while  the  ternary  ones  were  con- 
sidered independently  by  C.  Jordan  (iSSo)  and  H.  Valentiner  (1889). 
The  latter  discovered  the  important  group  of  order  360  which  was 
omitted  by  Jordan  and  has  recently  been  proved  (by  A.  Wiman  of 
Lund)  simply  isomorphic  to  the  alternating  group  of  degree  6.  H. 
Maschke  has  considered  many  quaternary  groups  and  established, 
in  particular,  a  complete  form  system  of  the  quaternary  group  of 
51840  linear  substitutions."  Heinrich  Maschke  (1853-1908)  was  born 
in  Breslau,  studied  in  Berlin  under  K.  Weierstrass,  E.  E.  Kummer, 
and  L.  Kronecker,  later  in  Gottingen  under  H.  A.  Schwarz,  J.  B. 
Listing,  and  F.  Klein.  He  entered  upon  the  study  of  group  theory 
under  Klein.  In  1891  he  came  to  the  United  States,  worked  a  year 
with  the  Weston  Electric  Co.,  then  accepted  a  place  at  the  University 
of  Chicago. 

Linear  groups  of  finite  order,  first  treated  by  Felix  Klein,  were 
later  used  by  him  in  the  extension  of  the  Galois  theory  of  algebraic 
equations,  as  seen  in  his  Ikosaeder.    As  stated  above,  Klein's  de- 


36o  A  HISTORY  OF  MATHEMATICS 

termination  of  the  linear  groups  in  two  variables  was  followed  by 
groups  in  three  variables,  developed  by  C.  Jordan  and  H.  Valentiner 
(1889),  and  by  groups  of  any  number  of  variables,  treated  by  C. 
Jordan.  Special  linear  groups  in  four  variables  were  discussed  by  E. 
Goursat  (1SS9)  and  G.  Bagncra  of  Palermo  (1905).  The  complete 
determination  of  the  groups  in  four  variables,  aside  from  intransitive 
and  monomial  types,  was  carried  through  by  H.  F.  Blichfeldt  of  Le- 
land  Stanford  University.^  Says  Blichfeldt:  "There  are,  in  the  main, 
four  distinct  principles  employed  in  the  determination  of  the  groups 
in  2,  3  or  4  variables:  (a)  the  original  geometrical  process  of  Klein  .  .  .  ; 
(b)  the  processes  leading  to  a  diophantine  equation,  which  may  be 
approached  analytically  (C.  Jordan  .  .  .),  or  geometrically  (H.  Valen- 
tiner, G.  Bagnera,  H.  H.  Mitchell);  (c)  a  process  involving  the  relative 
geometrical  properties  of  transformations  which  represent  'homologies' 
and  like  forms  (H.  Valentiner,  G.  Bagnera,  H.  H.  Mitchell  .  .  .  ;  (d) 
a  process  developed  from  the  properties  of  the  multipliers  of  the  trans- 
formations, which  are  roots  of  unity  (H.  F.  Blichfeldt).  A  new  prin- 
ciple has  been  added  recently  by  L.  Bieberbach,  though  it  had  already 
been  used  by  H.  Valentiner  in  a  certain  form.  .  .  .  Independent  of 
these  principles  stands  the  theory  of  group  characteristics,  of  which 
G.  Frobenius  is  the  discoverer." 

There  is  a  marked  difference  between  finite  groups  of  even  and  of 
odd  order.^  As  W.  Burnside  points  out,  the  latter  admit  no  self- 
inverse  irreducible  representation,  except  the  identical  one;  all  irre- 
ducible groups  of  odd  order  in  3,  5  or  7  symbols  are  soluble.  G.  A. 
Miller  proved  in  1901  that  no  group  of  odd  order  with  a  conjugate 
set  of  operations  containing  fewer  than  50  members  could  be  simple. 
W.  Burnside  proved  in  1901  that  transitive  groups  of  odd  order  whose 
degree  is  less  than  100  are  soluble.  H.  L.  Rietz  in  1904  extended  this 
last  result  to  groups  whose  degrees  are  less  than  243.  W.  Burnside 
has  shown  that  the  number  of  prime  factors  in  the  order  of  a  simple 
group  of  odd  order  cannot  be  less  than  7  and  that  40,000  is  a  lower 
limit  for  the  order  of  a  group  of  odd  degree,  if  simple.  These  results 
suggest  that,  perhaps,  simple  groups  of  odd  order  do  not  exist.  Recent 
researches  on  groups,  mainl}-  abstract  groups,  are  due  to  L.  E.  Dickson, 
Le  Vavasseur,  M.  Potron,  L.  I.  Neikirk,  G.  Frobenius,  H.  Hilton, 
A.  Wiman,  J.  A.  de  Seguicr,  H.  W.  Kuhn,  A.  Loewy,  H.  F.  Blichfeldt,^ 
W.  A.  Manning,  and  many  others.  Extensive  researches  on  abstract 
groups  have  been  carried  on  by  G.  A.  Miller  of  the  University  of 
Illinois.  In  1914  he  showed,  for  instance,  that  a  non-abelian  group 
can  have  an  abelian  group  of  isomorphisms  by  proving  the  existence 

'  We  are  using  H.  F.  Blichfeldt's  Finite  Collineation  Groups,  Chicago,  19 17, 
pp.  174-177- 

2  W.  Burnside,  Theory  of  Groups  of  Finite  Order,  2.  Ed.,  Cambridge,  1911,  p.  503. 

^  Consult  G.  A.  Miller's  "Third  Report  on  Recent  Progress  in  the  Theory  of 
Groups  of  Finite  Order"  in  Bull.  Am.  Math.  Soc,  Vol.  14,  1907,  p.  124- 


ALGEBRA  361 

of  this  relation  in  a  group  of  order  64.  He  proved  the  existence  of  a 
group  of  order  /»^,  p  being  any  prime  number  whatever,  whose  group 
of  isomorphisms  has  an  order  which  is  a  power  ^  of  p.  He  proved  also 
the  existence  of  a  group  G  of  order  128  which  admits  of  an  outer 
isomoq>hism  which  changes  each  conjugate  set  of  operations  into 
itself.  Among  other  results  due  to  G.  A.  Miller  are  these:  The  number 
of  independent  generators  of  every  prime  power  group  is  an  invariant 
of  the  group;  a  necessary  and  sufficient  condition  that  a  solvable 
group  is  a  direct  product  of  a  Sylow  subgroup  and  another  subgroup 
is  that  its  group  of  inner  isomorphisms  involves  the  corresponding 
Sylow  subgroup  as  a  factor  of  a  direct  product,  whenever  it  involves 
such  a  subgrouji.- 

A  work  which  embodied  modern  researches  in  algebra  was  the 
Lchrbiich  der  Algebra,  issued  by  H.  Weber  in  1895-1896  in  two  volumes, 
and  in  three  volumes  in  the  revised  edition  of  i8g8  and  1899.  Hein- 
rich  Weber  (1842-1913)  was  born  in  Heidelberg  and  studied  at 
Heidelberg,  Leipzig,  and  Konigsberg.  Since  1869  he  was  successively 
professor  at  Heidelberg,  Konigsberg,  Berlin,  Marburg,  Gottingen 
and  (since  1895)  at  Strassburg.  He  was  editor  of  Riemann's  Collected 
Works  (1876;  2.  ed.  1892).  He  carried  on  researches  in  algebra,  theory 
of  numbers,  theory  of  functions,  mechanics  and  mathematical  physics. 
In  191 1  he  mourned  the  loss  of  a  gifted  daughter  who  had  trans- 
lated Poincare's  Valeur  de  la  science  and  other  French  books  into 
German. 

The  symmetric  functions  of  the  sums  of  powers  of  the  roots  of  an 
equation,  studied  by  I.  Newton  and  E.  Waring,  was  considered  more 
recently  by  K.  F.  Gauss,  A.  Cayley,  J.  J.  Sylvester,  and  F.  Brioschi. 
Cayley  gives  rules  for  the  "weight"  and  "order"  of  symmetric  func- 
tions. 

The  theory  of  elimination  was  greatly  advanced  by  J.  J.  Sylvester, 
A.  Cayley,  G.  Salmon,  C.  G.  J.  Jacobi,  L.  O.  Hesse,  A.  L.  Cauchy, 
E.  Brioschi,  and  P.  Gordan.  Sylvester  gave  the  dialytic  method 
{Philosophical  Magazine,  1S40),  and  in  1852  established  a  theorem 
relating  to  the  expression  of  an  eliminant  as  a  determinant.  A.  Cayley 
made  a  new  statement  of  Bezout's  method  of  elimination  and  estab- 
lished a  general  theory  of  elimination  (1852). 

Contributions  to  the  theory  of  equations,  based  on  Descartes'  rule 
of  signs  and  especially  on  its  application  to  infinite  series  were  made 
by  Edmond  Laguerre  (1834-1886),  professor  in  the  College  de  France 
in  Paris.  An  upper  limit  for  the  number  of  real  roots  of  a  polynomial 
with  real  coefficients,  /(.r),  in  an  interval  {0,  a)  results  from  the  ap- 
plication of  the  rule  of  signs  to  a  product /"2(.r)=/i(x)  /(x)  developed 
in  a  power  series  which  converges  for  |  x  \<a,  but  diverges  for  .T=a. 
In  particular,  he  proved  that  if  z  in  e^^f(x)  is  taken  sufficiently  large, 

^  Bidl.  Am.  Math.  Soc,  Vol.  20,  1914,  pp.  310,  311. 
^Ibid,  Vol.  18,  1912,  p.  440. 


362  A  HISTORY  OF  MATHEMATICS 

then  the  exact  number  of  positive  roots  is  ascertainable  from  the 
variations  of  sign  in  the  series.  Michel  Fekete  and  Georg  Polya, 
both  of  Budapest,  use/(.T)/(i— x)"  for  the  same  purpose.^ 

The  theory  of  equations  commanded  the  attention  of  Leopold 
Kronecker  (1823-1S91).  He  was  born  in  Liegnitz  near  Breslau, 
studied  at  the  gymnasium  of  his  native  town  under  Kummer,  later 
in  BerHn  under  C.  G.  J.  Jacobi,  J.  Steiner,  and  P.  Dirichlet,  then  in 
Breslau  again  under  E.  E.  Kummer.  Though  for  eleven  years  after 
1844  engaged  in  business  and  the  care  of  his  estates,  he  did  not  neglect 
mathematics,  and  his  fame  grew  apace.  In  1855  he  went  to  Berlin 
where  he  began  to  lecture  at  the  University  in  186 1.  He  was  a  very 
stimulating  and  interesting  lecturer.  Kummer,  K.  Weierstrass,  and 
L.  Kronecker  constitute  the  triumvirate  of  the  second  mathematical 
school  in  Berlin.  This  school  emphasized  severe  rigor  in  demonstra- 
tions. L.  Kronecker  dwelt  intensely  upon  arithmetization  which 
repressed  as  far  as  possible  all  space  representations  and  rested  solely 
upon  the  concept  of  number,  particularly  the  positive  integer.  He 
displayed  manysided  talent  and  extraordinary  ability  to  penetrate 
new  fields  of  thought.  "But,"  says  G.  Frobenius,^  "conspicuous  as 
his  achievements  are  in  the  different  fields  of  number  research,  he 
does  not  quite  reach  up  to  A.  L.  Cauchy  and  C.  G.  J.  Jacobi  in  analy- 
sis, nor  to  B.  Riemann  and  Weierstrass  in  function-theory,  nor  to 
Dirichlet  and  Kummer  in  number- theory."  Kronecker's  papers  on 
algebra,  the  theory  of  equations  and  elliptic  functions  proved  to  be 
difficult  reading.  A  more  complete  and  simplified  exposition  of  his 
results  was  given  by  R.  Dedekind  and  H.  Weber.  "Among  the  finest 
of  Kronecker's  achievements,"  says  Fine,^  "were  the  connections 
which  he  established  among  the  various  disciplines  in  which  he  worked: 
notably  that  between  the  theory  of  quadratic  forms  of  negative  deter- 
minant and  elliptic  functions,  throu,^h  the  singular  moduli  which  give 
rise  to  the  complex  multiplication  of  the  elliptic  functions,  and  that 
between  the  theory  of  numbers  and  algebra,  by  his  arithmetical 
theory  of  the  algebraic  equation."  He  held  to  the  view  that  the  theory 
of  fractional  and  irrational  numbers  could  be  built  upon  the  integral 
numbers  alone.  "Die  ganze  zahl,"  said  he,  "schuf  der  liebe  Gott, 
alles  Uebrige  ist  Menschenwerk."  Later  he  even  denied  the  existence 
of  irrational  numbers.  He  once  paradoxically  remarked  to  Linde- 
mann:  "Of  what  use  is  your  beautiful  research  on  the  number  x? 
Why  cogitate  over  such  problems,  when  really  there  are  no  irrational 
numbers  whatever?  " 

In  1890-1891  L.  Kronecker  developed  a  theory  of  the  algebraic  equa- 
tion with  numerical  coefficients,  which  he  did  not  live  to  publish. 
From  notes  of  Kronecker's  lectures,  H.  B.  Fine  of  Princeton  prepared 

'  Bull.  Am.  Math.  Soc,  Vol.  20,  1913,  p.  20. 

*  G.  Frobenius,  GecTdchtnissrede  auf  Leopold  Kronecker,  Berlin,  1893,  p.  i. 

s  Bull.  Am.  Math.  Soc,  Vol  I;  1892,  p.  175. 


ALGEBRA  363 

an  address  m  1913  g"'ing  Kronecker's  unpublished  results.^  "All 
who  ha\'e  read  Kronecker's  later  writings,"  says  Fine,  "are  familiar 
with  his  contention  that  the  theory  of  the  algebraic  equation  in  its 
final  form  must  be  based  solely  on  the  rational  integer,  algebraic 
numbers  being  excluded  and  only  such  relations  and  operations  being 
admitted  as  can  be  expressed  in  finite  terms  by  means  of  rational 
numbers  and  therefore  ultimately  by  means  of  integers.  These  lec- 
tures of  1890-91  are  chiefly  concenied  with  the  development  of  such 
a  theory,  and  in  ]iarticular  with  the  proof  of  two  theorems  which 
therein  take  the  place  of  the  fundamental  theorem  of  algebra  as 
commonly  stated." 

Solution  of  Numerical  Equations 

Jacques  Charles  Franqois  Sturm  (1803-1855),  a  native  of  Geneva, 
Switzerland,  and  the  successor  of  Poisson  in  the  chair  of  mechanics 
at  the  Sorbonne,  published  in  1829  his  celebrated  theorem  determining 
the  number  and  situation  of  real  roots  of  an  equation  comprised 
between  given  limits.  De  Morgan  has  said  that  this  theorem  "is  the 
complete  theoretical  solution  of  a  difficulty  upon  which  energies  of 
every  order  have  been  employed  since  the  time  of  Descartes."  Sturm 
explains  in  that  article  that  he  enjoyed  the  privilege  of  reading  Four- 
ier's researches  while  they  were  still  in  manuscript  and  that  his  own 
discovery  was  the  result  of  the  close  study  of  the  principles  set  forth 
by  Fourier.  In  1829  Sturm  published  no  proof.  Proofs  were  given 
in  1830  by  Andreas  von  Ettinghausen  (1796-1878)  of  Vienna,  in 
1832  by  Charles  Choquet  et  Mathias  Mayer  in  their  Algebre,  and  in 
1S35  by  Sturm  himself.  According  to  J.  M.  C.  Duhamel,  Sturm's 
discovery  was  not  the  result  of  observation,  but  of  a  well-ordered 
line  of  thought  as  to  the  kind  of  function  that  would  meet  the  re- 
quirements. According  to  J.  J.  Sylvester,  the  theorem  "stared  him 
(Sturm)  in  the  face  in  the  midst  of  some  mechanical  investigations 
connected  with  the  motion  of  compound  pendulums."  Duhamel  and 
Sylvester  both  state  that  they  received  their  information  from  Sturm 
directly.  Yet  their  statements  do  not  agree.  Perhaps  both  statements 
are  correct,  but  represent  different  stages  in  the  evolution  of  the  dis- 
covery in  Sturm's  mind.' 

By  the  theorem  of  Sturm  one  can  ascertain  the  number  of  complex 
roots,  but  not  their  location.  That  limitation  was  removed  in  a  bril- 
liant research  by  another  great  Frenchman,  A.  L.  Cauchy.  He  dis- 
covered in  183 1  a  general  theorem  which  reveals  the  number  of  roots, 
whether  real  or  complex,  which  lie  within  a  given  contour.  This 
theorem  makes  heavier  demands  upon  the  mathematical  attainments 

1  Bull.  Am.  Math.  Soc,  Vol.  20,  191^,  p.  339. 

-Consult  also  M.  B6cher,  "The  published  and  unpublished  Work  of  Charles 
Sturm  on  algebraic  and  differential  Equations"  in  Bull.  Am.  Math.  Soc,  Vol.  18, 
1912,  pp.  1-18. 


364  A  HISTORY  OF  MATHEMATICS 

of  the  reader,  and  for  that  reason  has  not  the  celebrity  of  Sturm's 
theorem.  But  it  enlisted  the  lively  interest  of  men  like  Sturm,  J. 
Liouville,  and  F.  Moicjno. 

A  remarkable  article  was  published  in  1826  by  Germinal  Dandelin 
(1794-1S47)  in  the  memoirs  of  the  Academy  of  Sciences  of  Brussels. 
He  gave  the  conditions  under  which  the  Newton-Raphson  method  of 
approximation  can  be  used  with  security.  In  this  part  of  his  research 
he  was  anticipated  by  both  Mourraille  and  J.  Fourier.  In  another 
part  of  his  paper  (the  second  supplement)  he  is  more  fortunate;  there 
he  describes  a  new  and  masterly  device  for  approximating  to  the  roots 
of  an  equation,  which  constitutes  an  anticipation  of  the  famous 
method  of  C.  H.  Graffe.  We  must  add  here  that  the  fundamental 
idea  of  Graffe's  method  is  found  even  earlier,  in  the  Miscellanea 
analytica,  1762,  of  Edward  Waring.  If  a  root  lies  between  a  and  b, 
a—b<i,  and  a  is  on  the  convex  side  of  the  curve,  then  Dandelin 
puts  x=a+y  and  transforms  the  equation  into  one  whose  root  y  is 
small.  He  then  multipUes /(y)  by/(-y)  and  obtains,  upon  writing 
y^=z,  an  equation  of  the  same  degree  as  the  original  one,  but  whose 
roots  are  the  squares  of  the  roots  of  the  equation /(y)  =0.  He  remarks 
that  this  transformation  may  be  repeated,  so  as  to  get  the  fourth, 
eighth,  and  higher  powers,  whereby  the  moduli  of  the  powers  of  the 
roots  diverge  sufficiently  to  make  the  transformed  equation  separable 
into  as  many  polygons  as  there  are  roots  of  distinct  moduli.  He  ex- 
plains how  the  real  and  imaginary  roots  can  be  obtained.  Dandelin's 
research  had  the  misfortune  of  being  buried  in  the  ponderous  tomes 
of  a  royal  academy.  Only  accidentally  did  we  come  upon  this  antici- 
pation of  the  method  of  C.  H.  Graffe.  Later  the  Academy  of  Sciences 
£>i  Berlin  offered  a  prize  for  the  invention  of  a  practical  method  of 
computing  imaginary  roots.  The  prize  was  awarded  to  Carl  Heinrich 
Graffe  (1799-1873),  professor  of  mathematics  in  Zurich,  for  his  paper, 
published  in  1837  in  Zurich,  entitled.  Die  Auflosung  der  hoheren 
numerischen  Gleichungen.  This  contains  the  famous  "  Graffe  method," 
to  which  reference  has  been  made.  Graffe  proceeds  from  the  same 
principle  as  did  Moritz  Abraham  Stern  (1807-1894),  of  Gottingen  in 
the  method  of  recurrent  series,  and  as  did  Dandelin.  By  the  process 
of  involution  to  higher  and  higher  powers,  the  smaller  roots  are  caused 
to  vanish  in  comparison  to  the  larger.  The  law  by  which  the  new 
equations  are  constructed  is  exceedingly  simple.  If,  for  example, 
the  coefficient  of  the  fourth  term  of  the  given  equation  is  as,  then 
the  corresponding  coefficient  of  the  first  transformed  equation  is 
Oj  — 2a2'i4+2Ciao— 2a6.  In  the  computation  of  the  new  coefficients, 
Graffe  uses  logarithms.  By  this  remarkable  method  all  the  roots, 
both  real  and  imaginary,  are  found  simultaneously,  without  the 
necessity  of  determining  beforehand  the  number  of  real  roots  and 
the  location  of  each  root.  The  discussion  of  the  case  of  equal  imaginary 
roots,  omitted  by  Graffe^  was  taken  up  by  the  astronomer  J.  F.  Encke 


ALGEBRA  365 

'•■"1  1S41.  A  simplified  exposition  of  the  Dandelin-Graffe  method  was 
given  by  Emmanuel  Carvallo  in  1896;  it  resembles  in  some  parts  that 
of  Dandelin,  although  Carvallo  had  not  seen  Dandelin's  paper.  For 
didactic  purposes,  an  able  explanation  is  given  in  Gustav  Bauer's 
Vorlesiingen  ilber  Algebra,  1903. 

In  i860  E.  Fiirstenau  expressed  any  definite  real  root  of  an  alge- 
braic equation  with  numerical  or  literal  coefficients,  in  terms  of  its 
coefficients,  through  the  aid  of  infinite  determinants,  a  kind  of  de- 
terminant then  used  for  the  first  time.  In  1867  he  extended  his  results 
to  imaginary  roots.  The  approximation  is  made  to  depend  upon  the 
fact  used  by  Daniel  Bernoulli,  L.  Euler,  J.  Fourier,  M.  A.  Stern,  G. 
Dandelin,  and  C.  H.  GralTe,  that  high  powers  of  the  smaller  roots  are 
negligible  in  comparison  with  high  powers  of  the  greater  roots.  E. 
Fiirstenau's  process  was  elaborated  by  E.  Schroder  (1870),  Siegmund 
Giinther,  (1874),  and  Hans  Naegelsbach  (1876). 

Worthy  of  notice  is  "Weddle's  method"  of  solving  numerical 
equations,  devised  by  Thomas  Weddle  (1817-1853)  of  Newcastle  in 
England,  in  1S42.  It  is  kindred  to  that  of  W.  G.  Horner.  The  suc- 
cessive approximations  are  effected  by  multiplications  instead  of 
additions.  The  method  is  advantageous  when  the  degree  of  the 
equation  is  high  and  some  of  the  terms  are  missing.  It  has  received 
some  attention  in  Italy  and  Germany.  In  1851  Simon  Spitzer  ex- 
tended it  to  the  computation  of  complex  roots. 

The  solution  of  equations  by  infinite  series  which  was  a  favorite 
subject  of  research  during  the  eighteenth  century  (Thomas  Simpson, 
L.  Euler,  J.  Lagrange,  and  others),  received  considerable  attention 
during  the  nineteenth.  Among  the  early  workers  were  C.  G.  J. 
Jacobi  (1S30),  W.  S.  B.  Woolhouse  (1868),  O.  Schlomilch  (1849), 
but  none  of  their  devices  were  satisfactory  to  the  practical  computer. 
Later  writers  aimed  at  the  simultaneous  calculation  of  all  the  roots 
by  infinite  series.  This  was  achieved  for  a  three-term  equation  by 
R.  Dietrich  in  1883  and  by  P.  Nekrasofif  in  1887.  For  the  general 
equation  it  was  accomplished  in  1895  by  Emory  McCIintock  (1840- 
19 16),  an  actuary  in  New  York,  who  was  president  of  the  American 
Mathematical  Society  from  1890  to  1894.  He  used  a  series  derived 
by  his  Calculus  of  Enlargement,  but  which  may  be  derived  also  by 
applying  "Lagrange's  series."  A  prominent  part  in  McClintock's 
treatment  is  his  theory  of  "dominant"  coefficients,  which  theory  lacks 
precision,  inasmuch  as  no  criterion  is  given  to  ascertain  whether  a 
coefficient  is  dominant  or  not,  which  is  both  necessary  and  sufficient. 
Preston  A.  Lambert  of  Lehigh  University  used  Maclaurin's  series  in 
1903;  in  1908  he  paid  special  attention  to  convergency  conditions, 
pointing  out  that  the  conditions  for  a  /-term  equation  can  be  set  up 
when  those  of  a  (/  -  i)-term  equation  are  known.  In  Italy  Lambert's 
papers  were  studied  in  1906  by  C.  Rossi  and  in  1907  by  Alfredo 
Capclli  (1855-1910)  of  Naples.    These  recent  researches  of  American 


366  A  HISTORY  OF  MATHEMATICS 

and  Italian  mathematicians  have  placed  the  determination  of  real 
and  imaginary  roots  of  numerical  equations  by  the  method  of  infinite 
series  within  reach  of  the  practical  computer.  The  methods  them- 
selves indicate  the  number  of  real  and  imaginary  roots,  so  that  one 
can  dispense  with  the  application  of  Sturm's  theorem  here  just  as 
easily  as  one  can  in  the  Dandelin-Griiffe  method.  Considerable  at- 
tention has  been  given  to  the  solution  of  special  types — trinomial 
equations — by  G.  Dandelin  (1826),  K.  F.  Gauss  (1840,  1843),  J. 
Bellavitis  (1846),  Lord  John  M'Laren  (1890).  The  last  three  used 
logarithms  of  sums  and  differences,  which  were  first  suggested  by 
G.  Z.  LeonelH  in  1802  and  are  often  called  "Gaussian  logarithms." 
The  extension  of  the  Gaussian  method  to  quadrinomials  was  under- 
taken by  S.  Gundelfinger  in  1884  and  1885,  Carl  Faerber  in  1889,  and 
Alfred  Wiener  in  1886.  The  extension  of  the  Gaussian  method  to 
any  equation  was  taken  up  by  R.  Mehmke,  professor  in  Darmstadt, 
who  published  in  1889  a  logarithmic-graphic  method  of  solving  nu- 
merical equations,  and  in  1891  a  more  nearly  arithmetical  method  of 
solution  by  logarithms.  The  method  is  essentially  a  mixture  of  the 
Newton-Raphson  method  and  the  regula  falsi,  as  regards  its  theoretical 
basis.  Well  known  is  R.  Mehmke's  article  on  methods  of  computation 
in  the  Encyklopddie  der  mathematischen  Wissenschaften,  Vol.  i,  p.  938. 

Magic  Squares  and  Combinatory  Analysis 

The  latter  part  of  the  nineteenth  century  witnesses  a  revival  of 
interest  in  methods  of  constructing  magic  squares.  Chief  among  the 
writers  on  this  subject  are  J.  Horner  (1871),  S.  M.  Drach  (1873), 
Th.  Harmuth  (1S81),  W.  W.  R.  Ball  (1S93);  E.  Maillet  (1894),  E.  M. 
Laquiere  (18S0),  E.  Lucas  (1882),  E.  McClintock  (1897).^  Magic 
squares  of  the  "diabolic"  type,  as  Lucas  calls  them,  are  designated 
"  pandiagonal "  by  McClintock.  These  and  similar  forms  are  called 
"  Nasik  squares  "  by  A.  H.  Frost.  An  interesting  book,  Magic  Squares 
and  Cubes,  Chicago,  1908,  was  prepared  by  the  American  electrical 
engineer,  W.  S.  Andrews.  Still  more  recent  is  the  Combinatory  Anal- 
ysis, Vol.  I,  Cambridge,  1915,  Vol.  II,  1916,  by  P.  A.  MacMahon, 
which  touches  the  subject  of  magic  squares.  Says  MacMahon:  "In 
fact,  the  whole  subject  of  Magic  Squares  and  connected  arrange- 
ments of  numbers  appears  at  first  sight  to  occupy  a  position  which  is 
completely  isolated  from  other  departments  of  pure  mathematics. 
The  object  of  Chapters  II  and  III  is  to  establish  connecting  links 
where  none  previously  existed.  This  is  accomplished  by  selecting 
a  certain  differential  operation  and  a  certain  algebraical  function," 
I,  p.  VIII. 

"The  'Probleme  des  Rencontres'  .  .  .  can  be  discussed  in  the  same 
manner.  The  reader  will  be  familiar  with  the  old  question  of  the 
'  EncydopSdie  des  sciences  mal/iSm.  T.  I,  Vo\  2,  1906,  pp.  67-75. 


ANALYSIS  367 

letters  and  envelopes.  A  given  number  of  letters  are  written  to  dif- 
ferent persons  and  the  envelopes  correctly  addressed  but  the  letters 
are  placed  at  random  in  the  envelopes.  The  question  is  to  find  the 
probability  that  not  one  letter  is  put  into  the  right  envelope.  The 
enumeration  connected  with  this  probability  question  is  the  first 
step  that  must  be  taken  in  the  solution  of  the  famous  problem  of  the 
Latin  Square,"  I,  p.  IX. 

The  problem  of  the  Latin  Square:  "The  question  is  to  place  n  dif- 
ferent letters  a,  b,  c,  .  .  .  in  each  row  of  a  square  of  n'^  compartments 
in  such  wise  that,  one  letter  being  in  each  compartment,  each  column 
involves  the  whole  of  the  letters.  The  number  of  arrangements  is 
required.  The  question  is  famous  because,  from  the  time  of  Euler 
to  that  of  Cayley  inclusive,  its  solution  was  regarded  as  being  beyond 
the  powers  of  mathematical  analysis.  It  is  solved  without  difficulty 
by  the  method  of  dififerential  operators  of  which  we  are  speaking. 
In  fact  it  is  one  of  the  simplest  examples  of  the  method  which  is  shewn 
to  be  capable  of  solving  questions  of  a  much  more  recondite  charac- 
ter." ^ 

The  extension  of  the  principle  of  magic  squares  of  the  plane  to  three- 
dimensional  space  has  commanded  the  attention  of  many.  Most 
successful  in  this  field  were  the  Austrian  Jesuit  Adam  Adamandus 
Kochansky  (16S6).  the  Frenchman  Josef  Sauveur  (1710),  the  Germans 
Th.  Hugel,  (1S59)  and  Hermann  Schcffler  (1882). 

In  Vol.  II,  Major  MacMahon  gives  a  remarkable  group  of  identities 
discovered  by  S.  Ramanujan  of  Cambridge  which  have  applications 
in  the  partitions  of  numbers,  but  have  not  yet  been  established  by 
rigorous  demonstration. 

A  nalysis 

Under  this  head  we  find  it  convenient  to  consider  the  subjects  of 
the  differential  and  integral  calculus,  the  calculus  of  variations,  in- 
finite series,  probability,  differential  equations  and  integral  equations. 

An  early  representative  of  the  critical  and  philosophical  school  of 
mathematicians  of  the  nineteenth  century  was  Bernard  Bolzano 
(i 781-1848),  professor  of  the  philosophy  of  religion  at  Prague.  In 
1816  he  gave  a  proof  of  the  binomial  formula  and  exhibited  clear 
notions  on  the  convergence  of  series.  He  held  advanced  views  on 
variables,  continuity  and  limits.  He  was  a  forerunner  of  G.  Cantor. 
Noteworthy  is  his  posthumous  tract,  Paradoxien  des  Unendlichen 
(Preface,  1850),  edited  by  his  pupil,  Fr.  Pfihonsky.  Bolzano's  writings 
were  overlooked  by  mathematicians  until  H.  Hankel  called  attention 
to  them.  "He  has  everything,"  says  Hankel,  ''that  can  place  him 
in  this  respect  [notions  on  infinite  series]  on  the  same  level  with  Cauchy, 
only  not  the  art  peculiar  to  the  French  of  refining  their  ideas  and 
communicating  them  in  the  most  appropriate  and  taking  manner. 
1  P.  A.  MacMahon,  Combinaloty  Analysis,  Vol.  I,  Cambridge,  1915,  p.  ix. 


368  /v  HISTORY  OF  MATHEMATICS 

So  it  came  about  that  Bolzano  remained  unknown  and  was  soon  for 
gotten."  H.  A.  Schwarz  in  1872  looked  upon  Bolzano  as  the  inventor 
of  a  line  of  reasoning  further  developed  by  K.  Weierstrass.  In  1881 
O.  Stolz  declared  that  all  of  Bolzano's  writings  are  remarkable 
"inasmuch  as  they  start  with  an  unbiassed  and  acute  criticism  of  the 
contributions  of  the  older  literature."  ^ 

A  reformer  of  our  science  who  was  eminently  successful  in  reaching 
the  ear  of  his  contemporaries  was  Cauchy. 

Augustin-Louis  Cauchy^  (17S9-1857)  was  born  in  Paris,  and  re- 
ceived his  early  education  from  his  father.  J.  Lagrange  and  P.  S. 
Laplace,  with  whom  the  father  came  in  frequent^  contact,  foretold 
the  future  greatness  of  the  young  boy.  At  the  Ecole  Centrale  du 
Pantheon  he  excelled  in  ancient  classical  studies.  In  1805  he  entered 
the  Ecole  Polytechnique,  and  two  years  later  the  Ecole  des  Ponts  et 
Chaussees.  Cauchy  left  for  Cherbourg  in  iSio,  in  the  capacity  of 
engineer.  Laplace's  Mecanique  Celeste  and  Lagrange's  Fonctions 
Analytiqiies  were  among  his  book  companions  there.  Considerations 
of  health  induced  him  to  return  to  Paris  after  three  years.  Yielding 
to  the  persuasions  of  Lagrange  and  Laplace,  he  renounced  engineering 
in  fayor  of  pure  science.  We  find  him  next  holding  a  professorship  at 
the  Ecole  Polytechnique.  On  the  expulsion  of  Charles  X  and  the 
accession  to  the  throne  of  Louis  Philippe  in  1S30,  Cauchy,  being 
exceedingly  conscientious,  found  himself  unable  to  take  the  oath  de- 
manded of  him.  Being,  in  consequence,  deprived  of  his  positions,  he 
went  into  voluntary  exile.  At  Fribourg  in  Switzerland,  Cauchy  re 
sumed  his  studies,  and  in  183 1  was  induced  by  the  king  of  Piedmont 
to  accept  the  chair  of  mathematical  physics,  especially  created  for  him 
at  the  University  of  Turin.  In  1833  he  obeyed  the  call  of  his  exiled 
king,  Charles  X,  to  undertake  the  education  of  a  grandson,  the  Duke 
of  Bordeaux.  This  gave  Cauchy  an  opportunity  to  visit  various  parts 
of  Europe,  and  to  learn  how  extensively  his  works  were  being  read. 
Charles  X  bestowed  upon  him  the  title  of  Baron.  On  his  return  to 
Paris  in  1838,  a  chair  in  the  College  de  France  was  offered  to  him, 
but  the  oath  demanded  of  him  prevented  his  acceptance.  He  was 
nominated  member  of  the  Bureau  of  Longitude,  but  declared  ineligible 
by  the  ruling  power.  During  the  political  events  of  184S  the  oath  was 
suspended,  and  Cauchy  at  last  became  professor  at  the  Polytechnic 
School.  On  the  establishment  of  the  second  empire,  the  oath  was  re- 
instated, but  Cauchy  and  D.  F.  J.  Arago  were  exempt  from  it.  Cauchy 
was  a  man  of  great  piety,  and  in  two  of  his  publications  staunchly  de- 
fended the  Jesuits. 

Cauchy  was  a  prolific  and  profound  mathematician.  By  a  prompt 
publication  of  his  results,  and  the  preparation  of  standard  text-books, 
he  exercised  a  more  immediate  and  beneficial  influence  upon  tne  great 

'  Consult  H.  Bergman,  Das  Philosophischc  Werk  Bernard  Bolzanos,  Halle,  1909. 
^  C.  A.  Valson,  La  Vic  el  les  travaux  du  Baron  Cauchy,  Paris,  1868. 


ANALYSIS  369 

mass  of  mathematicians  than  any  contemporary  writer.  He  was  one 
of  the  leaders  in  infusing  rigor  into  analysis.  His  researches  extended 
over  the  field  of  series,  of  imaginaries,  theory  of  numbers,  differential 
equations,  theory  of  substitutions,  theory  of  functions,  determinants, 
mathematical  astronomy,  light,  elasticity,  etc., — covering  pretty 
much  the  whole  realm  of  mathematics,  pure  and  applied. 

Encouraged  by  P.  S.  Laplace  and  S.  D.  Poisson,  Cauchy  published  in 
1S21  his  Cours  (P  Analyse  de  VEcolc  Royale  Poly  technique,  a  work  of 
great  merit.  Had  it  been  studied  more  diligently  by  writers  of  text- 
books, many  a  lax  and  loose  method  of  analysis  long  prevalent  in  ele- 
mentary text-books  would  have  been  discarded  half  a  century  earlier. 
With  him  begins  the  process  of  "arithmetization."  He  made  the 
first  serious  attempt  to  give  a  rigorous  proof  of  Taylor's  theorem. 
He  greatly  improved  the  exposition  of  fundamental  principles  of  the 
differential  calculus  by  his  mode  of  considering  limits  and  his  new 
theory  on  the  continuity  of  functions.  Before  him,  the  limit  concept 
had  been  emphasized  in  France  by  D'Alembert,  in  England  by  I. 
Newton,  J.  Jurin,  B.  Robins,  and  C.  Maclaurin.  The  method  of 
Cauchy  was  accepted  with  favor  by  J.  M.  C.  Duhamel,  G.  J.  Houel, 
and  others.  In  England  special  attention  to  the  clear  exposition  of 
fundamental  principles  was  given  by  A.  De  Morgan  Cauchy  re- 
introduced the  concept  of  an  integral  of  a  function  as  the  limit  of  a 
sum,  a  concept  originally  due  to  G.  W.  Leibniz,  but  for  a  time  dis- 
placed by  L.  Euler's  integral  defined  as  the  result  of  reversing  differen- 
tiation. 

Calculus  of  Variations 

A.  L.  Cauchy  made  some  researches  on  the  calculus  of  variations. 
This  subject  had  long  remained  in  its  essential  principles  the  same  as 
when  it  came  from  the  hands  of  J.  Lagrange.  More  recent  studies  per- 
tain to  the  variation  of  a  double  integral  when  the  limits  are  also  vari- 
able, and  to  variations  of  multiple  integrals  in  general.  Memoirs  were 
published  by  K.  F.  Gauss  in  1829,  S.  D.  Poisson  in  183 1,  and  Michel 
Ostrogradski  (1801-1861)  of  St.  Petersburg  in  1834,  without,  however, 
determining  in  a  general  manner  the  number  and  form  of  the  equations 
which  must  subsist  at  the  limits  in  case  of  a  double  or  triple  integral. 
In  1837  C.  G.  J.  Jacobi  published  a  memoir,  showing  that  the  difficult 
integrations  demanded  by  the  discussion  of  the  second  variation,  by 
which  the  existence  of  a  maximum  or  minimum  can  be  ascertained, 
are  included  in  the  integrations  of  the  first  variation,  and  thus  are 
superfluous.  This  important  theorem,  presented  with  great  brevity 
by  C.  G.  J.  Jacobi,  was  elucidated  and  extended  by  V.  A.  Lebesgue, 
C.  E.  Delaunay,  Friedrich  Eisenlohr  '1831-1904),  Simon  Spitzer  (1826- 
1887)  of  Vienna,  L.  0.  Hesse,  and  R.  F.  A.  Clebsch.  A  memoir  by 
Pierre  P>ederic  Sarrus  (1708-1861)  of  the  University  of  Strasbourg  on 
ihe  question  of  determining  the  limiting  equations  which  must  be  com- 


370  A  HISTORY  OF  MATHEMATICS 

bined  with  the  indefinite  equations  in  order  to  determine  completely 
the  maxima  and  minima  of  multiple  integrals,  was  awarded  a  prize  by 
the  French  Academy  in  1845,  honorable  mention  being  made  of  a 
paper  by  C.  E.  Delaunay.  P.  F.  Sarrus's  method  was  simplified  by 
A.  L.  Cauchy.  In  1852  Gaspare  Mainardi  (1800-1879)  of  Pavia  at- 
tempted to  exhibit  a  new  method  of  discriminating  maxima  and 
minima,  and  extended  C.  G.  J.  Jacobi's  theorem  to  double  integrals. 
Mainardi  and  F.  Brioschi  showed  the  value  of  determinants  in  ex- 
hibiting the  terms  of  the  second  variation.  In  1861  Isaac  Todhimter 
(1820-1884)  of  St.  John's  College,  Cambridge,  published  his  valuable 
work  on  the  History  of  the  Progress  of  the  Calculus  of  Variations,  which 
contains  researches  of  his  own.  In  1866  he  published  a  most  important 
research,  developing  the  theory  of  discontinuous  solutions  (discussed 
in  particular  cases  by  A.  M.  Legendre),  and  doing  for  this  subject  what 
P.  F.  Sarrus  had  done  for  multiple  integrals. 

The  following  are  the  more  important  older  authors  of  systematic 
treatises  on  the  calculus  of  variations,  and  the  dates  of  publication: 
Robert  Woodhouse,  Fellow  of  Caius  College,  Cambridge,  1810; 
Richard  Abbatt  in  London,  1S37;  John  Hewitt  Jellett  (1S17-1888), 
once  Provost  of  Trinity  College,  Dublin,  1850;  Georg  Wilhelm  Strauch 
(1811-186S),  of  Aargau  in  Switzerland,  1849;  Frangois  Moigno  (1804- 
1884)  of  Paris,  and  Lorentz  Leonard  Lindelof  (1827-1908)  of  the 
University  of  Helsingfors,  in  1S61;  Lewis  Buffett  Carll  in  1881. 
Carll  (1844-1918),  was  a  bUnd  mathematician,  graduated  at  Co- 
lumbia College  in  1S70  and  in  1S91-1892  was  assistant  in  mathematics 
there. 

That,  of  all  plane  curves  of  given  length,  the  circle  includes  a  maxi- 
mum area,  and  of  all  closed  surfaces  of  given  area,  the  sphere  encloses 
a  maximum  volume,  are  theorems  considered  by  Archimedes  and 
Zenodorus,  but  not  proved  rigorously  for  two  thousand  years  until 
K.  Weierstrass  and  H.  A.  Schwarz.  Jakob  Steiner  thought  he  had 
proved  the  theorem  for  the  circle.  On  a  closed  plane  curve  different 
from  a  circle  four  non-cyclic  points  can  be  selected.  The  quadrilateral 
obtained  by  successively  joining  the  sides  has  its  area  increased  when 
it  is  so  deformed  (the  lunes  being  kept  rigid)  that  its  vertices  are 
cyclic.  Hence  the  total  area  is  increased,  and  the  circle  has  the  maxi- 
mum area.  Oskar  Perron  of  Tubingen  pointed  out  in  1913  by  an  ex- 
ample the  fallacy  of  this  proof:  Let  us  "prove"  that  i  is  the  largest  of 
all  positive  integers.  No  such  integer  larger  than  i  can  be  the  maxi- 
mum, for  the  reason  that  its  square  is  larger  than  itself.  Hence,  i 
must  be  the  maximum.  Steiner's  "proof"  does  not  prove  that  among 
all  closed  plane  curves  of  given  length  there  exists  one  whose  area  is  a 
maximum.^  K.  Weierstrass  gave  a  simple  general  existence  theorem 
applicable  to  the  extremes  of  continuous  (stetige)  functions.  The  max- 
imal property  of  the  sphere  was  first  proved  rigorously  in  1884  by 
'  See  W.  Blaschke  in  Jahresb.  d.  deutsch.  Math.  Vereinig.,  Vol.  24,  1915,  p.  195. 


ATTALYSIS  371 

H.  A.  Schwarz  by  the  aid  of  results  reached  by  K.  Weierstrass  in  the 
calculus  of  variations.  Another  proof,  based  on  geometrical  theorems, 
was  given  in  1901  by  Hermann  Minkowski. 

The  subject  of  minimal  surfaces,  which  had  received  the  attention  of 
J.  Lagrange,  A.  M.  Legendre,  K.  F.  Gauss  and  G.  Monge,  in  later 
time  commanded  the  special  attention  of  H.  A.  Schwarz.  The  blind 
physicist  of  the  University  of  Gand,  Joseph  Plateau  (1S01-1883),  in 
1S73  described  a  way  of  presenting  these  surfaces  to  the  eye  by  means 
of  soap  bubbles  made  of  glycerine  water.  Soap  bubbles  tend  to  be- 
come as  thick  as  possible  at  every  point  of  their  surface,  hence  to  make 
their  surfaces  as  small  as  possible.  More  recent  papers  on  minimal 
surfaces  are  by  Harris  Hancock  of  the  University  of  Cincinnati. 

Ernst  Pascal  of  the  University  of  Pavia  expressed  himself  in  1897 
on  the  calculus  of  variations  as  follows:^  "It  may  be  said  that  this  de- 
velopment [the  finding  of  the  differential  equatio  s  which  the  unknown 
functions  in  a  problem  must  satisfy]  closes  with  J.  Lagrange,  for  the 
later  analysts  turned  their  attention  chiefly  to  the  other,  more  dif- 
ficult problems  of  this  calculus.  The  problem  is  finally  disposed  of, 
if  one  considers  the  simplicity  of  the  formulas  which  arise;  wholly  dif- 
ferent is  this  matter,  if  one  considers  the  subject  from  the  standpoint 
of  rigor  of  derivation  of  the  formulas  and  the  extension  of  the  domain 
of  the  problems  to  which  these  formulas  are  applicable.  This  last 
is  what  has  been  done  for  some  years.  It  has  been  found  necessary  to 
prove  certain  theorems  which  underlie  those  formulas  and  which  the 
first  workers  looked  upon  as  axioms,  which  they  are  not."  This  new 
field  was  first  entered  by  I.  Todhunter,  M.  Ostrogradski,  C.  G.  J. 
Jacobi,  J.  Bertrand,  P.  du  Bois-Reymond,  G.  Erdmann,  R.  F.  A. 
Clebsch,  but  the  incisive  researches  which  mark  a  turning-point  in  the 
history  of  the  subject  are  due  to  K.  Weierstrass.  As  an  illustration 
of  Weierstrass's  method  of  communicatiing  many  of  his  mathematical 
results  to  others,  we  quote  the  following  from  0.  Bolza:  ^  "Unfortu- 
nately they  [results  on  the  calculus  of  variations]  were  g'ven  by  Weier- 
strass only  in  his  lectures  [since  1872I,  and  thus  became  known  only 
very  slowly  to  the  general  mathematical  public.  .  .  .  Weierstrass's 
results  and  methods  may  at  present  be  considered  as  generally  known, 
partly  through  dissertations  and  other  publications  of  his  pupils, 
partly  through  A.  Kneser's  Lehrbuch  der  Variationsrechnung  (Braun- 
schweig, 1900),  partly  through  sets  of  notes  ('Ausarbeitungen')  of 
which  a  great  number  are  in  circulation  and  copies  of  which  are  ac- 
cessible to  every  one  in  the  library  of  the  Mathematische  Verein 
at  BerUn,  and  in  the  Mathematische  Lesezimmer  at  Gottingen. 
Under  these  circumstances  I  have  not  hesitated  to  make  use  of  Weier- 
strass's lectures  just  as  if  they  had  been  published  in  print."  Weier- 
strass applied  modern  requirements  of  rigor  to  the  calculus  of  varia- 

1  E.  Pascal,  Die  Variationsrechnung,  iibers.  ■>•.  A.  Sclie[)p,  Leipzig,  1899,  p.  5 
''■  O.  Bolza,  Leclurcs  on  "ic  Calculm  of  Varialions,  Chicago,  1904,  pp.  l\,  xi. 


372  A  HISTORY  OF  MATHEMATICS 

tions  in  the  study  of  the  first  and  second  variation.  Not  only  did  he 
give  rigorous  proofs  for  the  first  three  necessary  conditions  and  for  the 
sufficiency  of  these  conditions  for  the  so-called  "weak"  extremum, 
but  he  also  extended  the  theory  of  the  first  and  second  variation  to 
the  case  where  the  curves  under  consideration  are  given  in  parameter 
representation.  He  discovered  the  fourth  necessary  condition  and  a 
sufficiency  proof  for  a  so-called  "strong"  extremum,  which  gave  for 
the  first  time  a  complete  solution  by  means  of  a  new  method,  based  on 
the  so-called  "  Weierstrass's  construction."  ^  Under  the  stimulus  of 
Weierstrass,  new  developments,  were  made  by  A.  Kneser,  then  of 
Dorpat,  whose  theory  is  based  on  the  extension  of  certain  theorems 
on  geodesies  to  extremals  in  general,  and  by  David  Hilbert  of  Got- 
tingen,  who  gave  an  "a  priori  existence  proof  for  an  extremum  of  a 
definite  integral — a  discovery  of  far-reaching  importance,  not  only  for 
the  Calculus  of  Variations,  but  also  for  the  theory  of  differential 
equations  and  the  theory  of  functions  "  (O.  Bolza).  In  190Q  Bolza 
published  an  enlarged  German  edition  of  his  calculus  of  variations, 
including  the  results  of  Gustav  v.  Escherich  of  Vienna,  the  Hilbert 
method  of  proving  Lagrange's  rule  of  multipliers  (multiplikator-regel), 
and  the  J.  W.  Lindeberg  of  Helsingfors  treatment  of  the  isoperimetric 
problem.  About  the  same  time  appeared  J.  Hadamard's  Calcul  des 
variations  recuellies  par  M.  Fruhet,  Paris,  1910.  Jacques  Hadamard 
(1865-  )  was  born  at  Versailles,  is  editor  of  the  Annales  scientifiques 
de  V (cole  normale  superieure.  In  1912  he  was  appointed  professor  of 
mathematical  analysis  at  the  Ecole  Poly  technique  of  Paris  as  successor 
to  Camille  Jordan.  In  the  above  mentioned  book  he  regards  the  cal- 
culus of  variations  as  a  part  of  a  new  and  broader  "  functional  calculus," 
along  the  lines  followed  also  by  V.  Volterra  in  his  functions  of  fines. 
This  functional  calculus  was  initiated  by  Maurice  Frechet  of  the  Uni- 
versity of  Poitiers  in  France.  The  authors  include  also  researches  by 
W.  F.  Osgood.  Other  prominent  researches  on  the  calculus  of  varia- 
tions are  due  to  J.  G.  Darboux,  E.  Goursat,  E.  Zermelo,  H.  A.  Schwarz, 
H.  Hahn,  anJ  to  the  Americans  H.  Hancock,  G.  A.  Bliss,  E.  R.  Hedrick, 
A.  L.  Underbill,  Max  Mason.  Bliss  and  Mason  systematically  ex- 
tended the  Weierstrassian  theory  of  the  calculus  of  variations  to 
problems  in  space. 

In  1858  David  Bier  ens  de  Haan  (182  2-1895)  of  Leiden  published  his 
Tables  d'Intcgrales  Dejlnies.  A  revision  and  the  consideration  of  the 
underlying  theory  appeared  in  1862.  It  contained  8339  formulas. 
A  critical  examination  of  the  latter,  made  by  E.  W.  Sheldon  in  1912, 
showed  that  it  was  "remarkably  free  from  error  when  one  imposes 
proper  limitations  upon  constants  and  functions,  not  stated  by  Haan. 

The  lectures  on  definite  integrals,  delivered  by  P.  G.  L.  Dirichlet 
in  1858,  were  elaborated  into  a  standard  work  in  1871  by  Gustav 
Ferdinand  Meyer  of  Munich. 

'  This  summary  is  taken  from  O.  Bolza,  op.  clt.,  Preface. 


ANALYSIS 


Convergence  of  Series 


373 


The  history  of  infinite  series  illustrates  vividly  the  salient  feature 
of  the  new  era  which  analysis  entered  upon  during  the  first  quarter 
of  this  century.  I.  Newton  and  G.  W.  Leibniz  felt  the  necessity  of 
inquiring  into  the  convergence  of  infinite  series,  but  they  had  no 
proper  criteria,  excepting  the  test  advanced  by  Leibniz  for  alternating 
series.  By  L.  Eulcr  and  his  contemporaries  the  formal  treatment  of 
series  was  greatly  extended,  while  the  necessity  for  determining  the 
convergence  was  generally  lost  sight  of.  L.  Euler  reached  some  very 
pretty  results  on  infinite  series,  now  well  kno^^Tl,  and  also  some  very 
absurd  results,  now  quite  forgotten.  The  faults  of  his  time  found 
their  culmination  in  the  Combinatorial  School  in  Germany,  which 
has  now  passed  into  oblivion.  This  combinatorial  school  was  founded 
by  Carl  Friedrich  Ilindcnbiirg  (1741-180S)  of  Leipzig  whose  pupils 
filled  many  of  the  German  University  chairs  during  the  first  decennium 
of  the  nineteenth  century.  The  first  important  and  strictly  rigorous 
investigation  of  infinite  series  was  made  by  K.  F.  Gauss  in  connection 
with  the  hypergeometric  series.  This  series,  thus  named  by  J.  Wallis, 
had  been  treated  by  L.  Euler  in  1769  and  177S  from  the  triple  stand- 
point of  a  power-series,  of  the  integral  of  a  certain  linear  differential 
equation  of  the  second  order,  and  of  a  definite  integral.  The  criterion 
developed  by  K.  F.  Gauss  settles  the  question  of  convergence  of  the 
hvpergeometric  series  in  every  case  which  it  is  intended  to  cover,  and 
thus  bears  the  stamp  of  generality  so  characteristic  of  Gauss's  writings. 
Owing  to  the  strangeness  of  treatment  and  unusual  rigor.  Gauss's 
paper  excited  little  interest  among  the  mathematicians  of  that  time. 

More  fortunate  in  reaching  the  public  was  A.  L.  Cauchy,  whose 
Analyse  Algcbriqiie  of  1821  contains  a  rigorous  treatment  of  series. 
All  series  whose  sum  does  not  approach  a  fixed  limit  as  the  number 
of  terms  increases  indefinitely  are  called  divergent.  Like  Gauss,  he 
institutes  comparisons  with  geometric  series,  and  finds  that  series 
with  positive  terms  are  convergent  or  not,  according  as  the  nXh.  root 
of  the  nth  term,  or  the  ratio  of  the  («-I-i)th  term  and  the  wth  term, 
is  ultimately  less  or  greater  than  unity.  To  reach  some  of  the  cases 
where  these  expressions  become  ultimately  unity  and  fail,  Cauchy 
established  two  other  tests.  He  showed  that  series  with  negative 
terms  converge  when  the  absolute  values  of  the  terms  converge,  and 
then  deduces  G.  W.  Leibniz's  test  for  alternating  series.  The  product 
of  two  convergent  series  was  not  found  to  be  necessarily  convergent. 
Cauchy's  theorem  that  the  product  of  two  absolutely  convergent 
scries  converges  to  the  product  of  the  sums  of  the  two  series  was 
5ho\\'n  half  a  century  later  by  F.  Mertens  of  Graz  to  be  still  true  if, 
of  the  two  convergent  series  to  be  multiplied  together,  only  one  is 
absolutely  convergent. 

The  most  outspoken  critic  of  the  old  methods  in  series  was  N.  H, 


374  A  HISTORY  OF  MATHEMATICS 

Abel.  His  letter  to  his  friend  B.  M.  Holmboe  (1826)  contains  severe 
criticisms.  It  is  very  interesting  reading,  even  to  modem  students. 
In  his  demonstration  of  the  binomial  theorem  he  established  the 
theorem  that  if  two  series  and  their  product  series  are  all  convergent, 
then  the  product  series  will  converge  towards  the  product  of  the 
sums  of  the  two  given  series.  This  remarkable  result  would  dispose 
of  the  whole  problem  of  multiplication  of  series  if  we  had  a  universal 
practical  criterion  of  convergency  for  semi-convergent  series.  Since 
we  do  not  possess  such  a  criterion,  theorems  have  been  recently  es- 
tablished by  A.  Pringsheim  of  Munich  and  A.  Voss  now  of  Munich 
which  remove  in  certain  cases  the  necessity  of  applying  tests  of  con- 
vergency to  the  product  series  by  the  application  of  tests  to  easier 
related  expressions.  A.  Pringsheim  reaches  the  following  interesting 
conclusions:  The  product  of  two  conditionally  convergent  series  can 
never  converge  absolutely,  but  a  conditionally  convergent  series,  or 
even  a  divergent  series,  multiplied  by  an  absolutely  convergent  series, 
may  yield  an  absolutely  convergent  product. 

The  researches  of  N.  H.  Abel  and  A.  L.  Cauchy  caused  a  considerable 
stir.  We  are  told  that  after  a  scientific  meeting  in  which  Cauchy 
had  presented  his  first  researches  on  series,  P.  S.  Laplace  hastened 
home  and  remained  there  in  seclusion  until  he  had  examined  the 
series  in  his  Mecanique  Celeste.  Luckily,  every  one  was  found  to  be 
convergent!  We  must  not  conclude,  however,  that  the  new  ideas 
at  once  displaced  the  old.  On  the  contrary,  the  new  views  were 
generally  accepted  only  after  a  long  struggle.  As  late  as  1844  A.  De 
Morgan  began  a  paper  on  "divergent  series"  in  this  style:  "I  believe 
it  will  be  generally  admitted  that  the  heading  of  this  paper  describes 
the  only  subject  yet  remaining,  of  an  elementary  character,  on  which 
a  serious  schism  exists  among  mathematicians  as  to  the  absolute 
correctness  or  incorrectness  of  results." 

First  in  time  in  the  evolution  of  more  delicate  criteria  of  convergence 
and  divergence  come  the  researches  of  Josef  Ludwig  Raabe  (1801- 
1859)  of  Zurich,  in  Crelle,  Vol.  IX;  then  follow  those  of  A.  De  Morgan 
as  given  in  his  calculus.  A.  De  Morgan  established  the  logarithmic 
criteria  which  were  discovered  in  part  independently  by  J,  Bertrand. 
The  forms  of  these  criteria,  as  given  by  J.  Bertrand  and  by  Ossian 
Bonnet,  are  more  convenient  than  De  Morgan's.  It  appears  from 
N.  H.  Abel's  posthumous  papers  that  he  had  anticipated  the  above- 
named  writers  in  establishing  logarithmic  criteria.  It  was  the  opin- 
ion of  Bonnet  that  the  logarithmic  criteria  never  fail;  but  P.  Du 
Bois-Reymond  and  A.  Pringsheim  have  each  discovered  series  demon- 
strably convergent  in  which  these  criteria  fail  to  determine  the  con- 
vergence. The  criteria  thus  far  alluded  to  have  been  called  by  Pring- 
sheim special  criteria,  because  they  all  depend  upon  a  comparison  of 
the  nth.  term  of  the  series  with  special  functions  a^,  n^,  n(\og  nY,  etc. 
Among  the  first  to  suggest  general  criteria,  and  to  consider  the  subject 


ANALYSIS  375 

from  a  still  wider  point  of  view,  culminating  in  a  regular  mathematical 
theor}-,  was  E.  E.  Kummer.  He  established  a  theorem  yielding  a  test 
consisting  of  two  parts,  the  first  part  of  which  was  afterwards  found 
to  be  superfluous.  The  study  of  general  criteria  was  continued  by 
Ulisse  Dini  (1S45-1918)  of  Pisa,  P.  Du  Bois-Reymond,  G.  Kohn  of 
Vienna,  and  A.  Pringsheim.  Du  Bois-Reymond  divides  criteria  into 
two  classes:  criteria  of  the  first  kind  and  criteria  of  the  second  kind,  ac- 
cording as  the  general  «th  term,  or  the  ratio  of  the  (n-l-i)th  term  and 
the  wth  term,  is  made  the  basis  of  research.  E.  E.  Rummer's  is  a 
criterion  of  the  second  kind.  A  criterion  of  the  first  kind,  analogous 
to  this,  was  invented  by  A.  Pringsheim.  From  the  general  criteria 
established  by  Du  Bois-Reymond  and  Pringsheim  respectively,  all 
the  special  criteria  can  be  derived.  The  theory  of  Pringsheim  is  very 
complete,  and  offers,  in  addition  to  the  criteria  of  the  first  kind  and 
second  kind,  entirely  new  criteria  of  a  third  kind,  and  also  generalized 
criteria  of  the  second  kind,  which  apply,  however,  only  to  series  with 
never  increasing  terms.  Those  of  the  third  kind  rest  mainly  on  the 
consideration  of  the  limit  of  the  difference  either  of  consecutive  terms 
or  of  their  reciprocals.  In  the  generalized  criteria  of  the  second  kind 
he  does  not  consider  the  ratio  of  two  consecutive  terms,  but  the  ratio 
of  any  two  terms  however  far  apart,  and  deduces,  among  others,  two 
criteria  previously  given  by  Gustav  Kohn  and  W.  Ermakoff  respec- 
tively. 

It  is  a  strange  vicissitude  that  divergent  series,  which  early  in  the 
nineteenth  century  were  supposed  to  have  been  banished  once  for 
all  from  rigorous  mathematics,  should  at  its  close  be  invited  to  return. 
In  1SS6  T.  J.  Stieltjes  and  H.  Poincare  showed  the  importance  to 
analysis  of  the  asymptotic  series,  at  that  time  employed  in  astronomy 
alone.  In  other  fields  of  research  G.  H.  Halphen,  E.  N.  Laguerre,  and 
T.  J.  Stieltjes  have  encountered  particular  examples  in  which,  a  whole 
series  being  divergent,  the  corresponding  continued  fraction  was 
convergent.  In  1894  H.  Pade  now  of  Bordeaux,  established  the  possi- 
bility of  defining,  in  certain  cases,  a  function  by  an  entire  divergent 
series.  This  subject  was  taken  up  also  by  J.  Hadamard  in  1892, 
C.  E.  Fabry  in  1896  and  M.  Ser\^ant  in  1899.  Researches  on  divergent 
series  have  been  carried  on  also  by  H.  Poincare,  E.  Borel,  T.  J.  Stieltjes, 
E.  Cesaro,  W.  B.  Ford  of  Michigan  and  R.  D.  Carmichael  of  Illi- 
nois. Thomas-Jean  Stieltjes  (1856-1894)  was  born  in  Zwolle  in 
Holland,  came  in  1882  under  the  influence  of  Ch.  Hermite,  became  a 
French  citizen,  and  later  received  a  professorship  at  the  University 
of  Toulouse.  Stieltjes  was  interested  not  only  in  divergent  and  con- 
ditionally convergent  series,  but  also  in  G.  F.  B.  Riemann's  ^  function 
and  the  theory  of  numbers. 

Difficult  questions  arose  in  the  study  of  Fourier's  series.^     A.  L. 

^  Arnold  Sachse,  Vcrsuch  einer  GeschicJitc  drr  Darslellung  willkurlichcr  Funk- 
iionen  einer  variahlen  durch  trlgonomelrischc  Rcihen,  Gottingen,  1879. 


376  A  HISTORY  OF  MATHEMATICS 

Cauchy  was  the  first  who  felt  the  necessity  of  inquiring  into  its  con- 
vergence. But  his  mode  of  proceeding  was  found  by  P.  G.  L.  Dirichlet 
to  be  unsatisfactory.  Dirichlet  made  the  first  thorough  researches 
on  this  subject  {Crclle,  Vol.  IV).  They  culminate  in  the  result  that 
whenever  the  function  does  not  become  infinite,  does  not  have  an 
infinite  number  of  discontinuities,  and  does  not  possess  an  infinite 
number  of  maxima  and  minima,  then  Fourier's  series  converges  toward 
the  value  of  that  function  at  all  places,  except  points  of  discontinuity, 
and  there  it  converges  toward  the  mean  of  the  two  boundary  values. 
L.  Schlafli  of  Bern  and  P.  Du  Bois-Reymond  expressed  doubts  as  to 
the  correctness  of  the  mean  value,  which  were,  however,  not  well 
founded.  Dirichlet's  conditions  are  sufficient,  but  not  necessary. 
Rudolf  Lipschitz  (1832-1903),  of  Bonn,  proved  that  Fourier's  series 
still  represents  the  function  when  the  number  of  discontinuities  is 
infinite,  and  established  a  condition  on  which  it  represents  a  function 
having  an  infinite  number  of  maxima  and  minima.  Dirichlet's  belief 
that  all  continuous  functions  can  be  represented  by  Fourier's  series 
at  all  points  was  shared  by  G.  F.  B.  Riemann  and  H.  Hank  el,  but 
was  proved  to  be  false  by  Du  Bois-Reymond  and  H.  A.  Schwarz. 
A.  Hurwitz  showed  how  to  express  the  product  of  two  ordinary  Fourier 
series  in  the  form  of  another  Fourier  series.  W.  W.  Kiistermann 
solved  the  analogous  problem  for  double  Fourier  series  in  which  a 
relation  involving  Fourier  constants  figures  vitally.  For  functions 
of  a  single  variable  an  analogous  relation  is  due  to  M.  A.  Parseval 
and  was  proved  by  him  under  certain  restrictions  on  the  nature  of 
convergence  of  the  Fourier  series  involved.  In  1893  de  la  Vallee 
Poussin  gave  a  proof  requiring  merely  that  the  function  and  its  square 
be  integrable.  A.  Hurwitz  in  1903  gave  further  developments.  More 
recently  the  subject  has  commanded  general  interest  through  the  re- 
searches of  Frigyes  Riesz  and  Ernst  Fischer  (Riesz-Fischer  theorem).^ 
Riemann  inquired  what  properties  a  function  must  have,  so  that 
there  may  be  a  trigonometric  series  which,  whenever  it  is  convergent, 
converges  toward  the  value  of  the  function.  He  found  necessary 
and  sufficient  conditions  for  this.  They  do  not  decide,  however, 
whether  such  a  series  actually  represents  the  function  or  not.  Rie- 
mann rejected  Cauchy's  definition  of  a  definite  integral  on  account  of 
its  arbitrariness,  gave  a  new  definition,  and  then  inquired  when  a 
function  has  an  integral.  His  researches  brought  to  light  the  fact 
that  continuous  functions  need  not  always  have  a  differential  coeffi- 
cient. But  this  property,  which  was  shown  by  K.  Weierstrass  to  be- 
long to  large  classes  of  functions,  was  not  found  necessarily  to  exclude 
them  from  being  represented  by  Fourier's  series.  Doubts  on  some  of 
the  conclusions  about  Fourier's  series  were  thrown  by  the  observation, 
made  by  Weierstrass,  that  the  integral  of  an  infinite  series  can  be 
shown  to  be  equal  to  the  sum  of  the  integrals  of  the  separate  terms 
'  Summary  taken  from  Bu^l.  Am.  Math.  Soc,  Vol.  22,  1915,  p.  6. 


ANALYSIS  377 

only  when  the  series  converges  uniformly  within  the  region  in  question. 
The  subject  of  uniform  convergence  was  first  investigated  in  1847  by 
G.  G.  Stokes  of  Cambridge  and  in  1S48  bv  Philip p  Ludwig  v.  Seidel 
(1821-1896).  !-eidel  had  studied  under  F.  W.  Bessel,  C.  G.  J.  Jacobi, 
J.  F.  Encke,  and  P.  G.  L.  Dirichlet.  He  became  professor  at  the 
University  of  Munich  in  1S55.  Later  his  lecturing  and  scientific 
activity  were  stopped  by  a  disease  of  his  eyes.  Uniform  convergence 
assumed  great  importance  in  K.  Weierstrass'  theory  of  functions.  It 
became  necessary  to  prove  that  a  trigonometric  series  representing 
a  continuous  function  converges  uniformly.  This  was  done  by  Hein- 
rich  Ediiard  Heine  (1821-1881),  of  Halle.  Later  researches  on  Four- 
ier's series  were  made  by  G.  Cantor  and  Paul  Du  Bois-Reyrnond 
(1831-1S89),  professor  at  the  technical  high  school  in  Charlottenburg. 
Less  stringent  than  that  of  uniform  convergence  is  U.  Dini's  def- 
inition ^  of  "simple  uniform  convergence,"  which  is  as  follows:  The 
series  is  said  to  be  simply  uniformly  convergent  in  the  interval  (a,  b) 
when  corresponding  to  every  arbitrarily  chosen  pofitive  number  a 
as  small  as  we  please  and  to  every  integer  m' ,  only  one  or  several 
integers  m  exist  which  are  not  less  than  r,i' ,  and  are  such  that,  for  all 
the  values  of  x  in  the  interval  (a,  b),  the  |  Rmi.^)  \  are  <5.  Still  another 
kind  of  convergence,  the  "uniform  convergence  by  segments,"  some- 
times called  "sub-uniform  convergence,"  was  introduced  in  18S3  by 
Cesare  Arzela  (1847-1912)  of  the  University  of  Bologna.  He  advanced 
the  theory  of  functions  of  real  variables  and  generalized  a  theorem  of 
U.  Dini  on  the  necessary  and  sufficient  conditions  for  the  continuity 
of  the  sum  of  a  convergent  series  of  continuous  functions. 

Probability  and  Siatistics 

As  compared  with  the  vast  development  of  other  mathematical 
branches,  the  theory  of  probability  has  made  insignificant  progress 
smce  the  time  of  P.  S.  Laplace.  Jakob  BernouUi's  Theorem  which 
had  received  the  careful  attention  of  De  JNIoivre,  J.  Stirling,  C.  Ma- 
claurin,  and  L.  Euler,  was  considered  especially  by  P.  S.  Laplace  who 
made  an  inverse  application  of  it,  assuming  that  an  event  had  been 
observed  to  happen  m  times  and  to  fail  n  times  in  n  trials,  and  then 
deducing  the  initially  unknown  probability  of  its  happening  at  each 
trial.  The  result  thus  obtained  did  not  agree  accurately  with  the 
results  gotten  by  the  use  of  Bayes'  Theorem.  The  subject  was  in- 
vestigated by  S.  D.  Poisson  in  his  Recherches  sur  la  probabilite,  Paris, 
1S37,  who  obtained  consonant  results  after  carrying  the  approxima- 
tions, in  the  use  of  Bayes'  Theorem,  to  a  higher  degree.  An  endeavor 
to  remove  the  obscurities  in  which  Bayes'  Theorem  seemed  involved 
was  made  by  Poisson,  and  by  A.  De  Morgan  in  his  Theory  of  Probabil- 

'  Grundlage:i  f.  c.  Theorle  dcr  Fimrlioncn,  by  J.  Liiroth  u.  A.  Schepp,  Leipzig, 
1892,  p.  137. 


378  A  HISTORY  OF  AIATHEMATICS 

ities,^  through  the  use  of  illustrations  with  urns  that  were  exactly 
alike  and  contained  black  and  white  balls  in  different  numbers  and 
different  ratios,  the  observed  event  being  the  drawing  of  a  white  ball 
from  any  one  of  the  urns.  For  the  same  purpose  Johannes  von  Kries, 
in  his  Prinzipien  der  Wahrscheinlichkeltsrechnung,  Freiburg  i.  B.,  iS86, 
used  as  an  illustration  six  equal  cubes,  of  which  one  had  the  +  sign 
on  one  side,  another  had  it  on  two  sides,  a  third  on  three  sides  and 
finally  the  sixth  on  all  six  sides.  All  other  sides  were  marked  with  a  o. 
Nevertheless,  objections  to  certain  applications  by  Bayes'  Theorem 
have  been  raised  by  the  Danish  actuary  J.  Bing  in  the  Tidsskrift  for 
Malematik,  1879,  by  Joseph  Bertrand  in  his  Calcid  des  probabilites, 
Paris,  1889,  by  Thorwald  Nicolai  Thiele  (1838-1910)  of  the  observa- 
tory at  Copenhagen  in  a  work  published  at  Copenhagen  in  1889  (an 
English  edition  of  which  appeared  under  the  title  Theory  of  Observa- 
tions, London,  1903)  by  George  Chrystal  (1S51-1911)  of  the  Univer- 
sity of  Edinburgh,  and  others.^  As  recently  as  1908  the  Danish 
philosophic  writer  Kroman  has  come  out  in  defence  of  Bayes.  Thus 
it  appears  that,  as  yet,  no  unanimity  of  judgment  has  been  reached 
in  this  matter.  In  determining  the  probabihty  of  alternative  causes 
deduced  from  observed  events  there  is  often  need  of  evidence  other 
than  that  which  is  afforded  by  the  observed  event.  By  inverse  prob- 
ability some  logicians  have  explained  induction.  For  example,  if  a 
man,  who  has  never  heard  of  the  tides,  were  to  go  to  the  shore  of  the 
Atlantic  Ocean  and  witness  on  m  successive  days  the  rise  of  the  sea, 
then,  says  Adolphe  Quetelet  of  the  observatory  at  Brussels,  he  would 

be  entitled  to  conclude  that  there  was  a  probability  equal  to 

that  the  sea  would  rise  next  day.  Putting  m=o,  it  is  seen  that  this 
view  rests  upon  the  unwarrantable  assumption  that  the  probabihty 
of  a  totally  unknown  event  is  ^,  or  that  of  all  theories  proposed  for 
investigation  one-half  are  true.  William  Stanley  Jevons  (1835-1882) 
in  his  Principles  of  Science  founds  induction  upon  the  theory  of  inverse 
probability,  and  F.  Y.  Edgeworth  also  accepts  it  in  his  Mathematical 
Psychics.  Daniel  BernoulU's  "moral  expectation,"  which  was  elab- 
orated also  by  Laplace,  has  received  little  attention  from  more  recent 
French  writers.  Bertrand  emphasizes  its  impracticability;  Poincare, 
in  his  Calcul  des  probabilites,  Paris,  1S96,  disposes  of  it  in  a  few  words.^ 
The  only  noteworthy  recent  addition  to  probability  is  the  subject 
of  "local  probability,"  developed  by  several  English  and  a  few  Amer- 
ican and  French  mathematicians.  G.  L.  L.  Buffon's  needle  problem, 
is  the  earliest  important  problem  on  local  probability;  it  received  the 

^  Encyclopedia  Metrop.  II,  1845. 

2  We  are  using  Emanuel  Czuber's  Entwick clung  dcr  Wahrschdnlichheitsiheorie  in 
the  Jahresb.  d.  deidsch.  Mathematiker-Vcrriiiignng,  i8qg,  pp.  Q3-105;  also  Arne 
Fisher,  The  Mathematical  Theory  of  Probabilities,  New  York,  IQ15,  pp.  54-56. 

^  E.  Czuber,  op.  cit.,  p.  121. 


ANALYSIS  379 

consideration  of  P.  S.  Laplace,  of  Emile  Barbier  in  the  years  i860 
and  \S82,  of  Morgan  W.  Crofton  (1S26-1915)  of  the  mihtary  school 
at  Woohvich,  who  in  1S6S  contributed  a  paper  to  the  London  Phil- 
osophical Transactions,  Vol.  158,  and  in  1885  wrote  the  article  "Prob- 
ability" in  the  Encyclopedia  Brittannica,  ninth  edition.  The  name 
''local  probability"  is  due  to  Crofton.  Through  considerations  of 
local  probability  he  was  led  to  the  evaluation  of  certain  definite  in- 
tegrals. 

Noteworthy  is  J.  J.  Sylvester's  four  point  problem:  To  find  the 
probability  that  four  points  taken  at  random  within  a  given  boundary, 
shall  form  a  re-entrant  quadrilateral.  Local  probability  was  studied 
in  England  also  by  A.  R.  Clarke,  H.  AIcCoU,  S.  Watson,  J.  Wolsten- 
holme,  W.  S.  Woolhouse;  in  France  also  by  C.  Jordan  and  E.  Lemoine; 
in  America  by  E.  B.  Seitz.  Rich  collections  of  problems  on  local  prob- 
ability have  been  published  by  Emanuel  Czuber  of  Vienna  in  his 
Geometrische  W ahrscheinlichkeikn  und  MiUelwerte,  Leipzig,  1884,  and 
by  G.  B.  M.  Zerr  in  the  Educational  Times,  Vol.  55,  1891,  pp.  137- 
192.  The  fundamental  concepts  of  local  probability  have  received  the 
special  attention  of  Ernesto  Cesaro  (1859-1906)  of  Naples.^ 

Criticisms  occasionally  passed  upon  the  principles  of  probability 
and  lack  of  confidence  in  theoretical  results  have  induced  several 
scientists  to  take  up  the  experimental  side,  which  had  been  emphasized 
by  G.  L.  L.  Buffon.  Trials  of  this  sort  were  made  by  A.  De  Morgan, 
W.  S.  Jevons,  L.  A.  J.  Quetelet,  E.  Czuber,  R.  Wolf,  and  showed  a 
remarkably  close  agreement  with  theory.  In  Buffon's  needle  prob- 
lem, the  theoretical  probability  involves  tt.  This  and  similar  expres- 
sions -  have  been  used  for  the  empirical  determination  of  tt.  Attempts 
to  place  the  theory  of  probability  on  a  purely  empirical  basis  were 
made  by  John  Stuart  Mill  (i 806-1 873),  John  Venn  (1834-  )  and 
G.  Chrj^stal.  Mill's  induction  method  was  put  on  a  sounder  basis 
by  A.  A.  Chuproff  in  a  brochure.  Die  Statistik  als  Wissenschaft.  Em- 
pirical methods  have  commanded  the  attention  of  another  Russian, 
V.  Bortkievicz. 

In  1835  and  1836  the  Paris  Academy  was  led  by  S.  D.  Poisson's 
researches  to  discuss  the  topic,  whether  questions  of  morality  could  be 
treated  by  the  theor}'  of  probability.  M.  H.  Navier  argued  on  the 
affirmative,  while  L.  Poinsot  and  Ch.  Dupin  denied  the  applicability 
as  "une  sorte  d'aberration  de  I'esprit;"  they  declared  the  theory 
applicable  only  to  cases  where  a  separation  and  counting  of  the  cases 
or  events  was  possible.  John  Stuart  Mill  opposed  it;  Joseph  L.  F. 
Bertrand  (1822-1900),  professor  at  the  College  de  France  in  Paris 
and  J.  v.  Kries  are  among  more  recent  writers  on  tliis  topic.^ 

^  See  Encydopidie  dts  sciences  math.  I,  20  {igo6),  p.  23. 
2  E.  Czuber,  op.  cil.,  pp.  88-91. 

■^  Consult  E.  Czuber,  op.  cit.,  p.  141;  J.  ?^.  Mill,  Sy.stcm  of  Logic,  New  York,  8lh 
Ld.,  1884,  Chap.  i3,  pp.  379-3^^7;  J-  V.  krics,  op.  c'U.,  pp.  253-259. 


38o  A  HISTORY  OF  MATHEMATICS 

Among  the  various  applications  of  probability  the  one  relating  to 
verdicts  of  juries,  decisions  of  courts  and  results  of  elections  is  specially 
interesting.  This  subject  was  studied  by  Marquis  de  Condorcet, 
P.  S.  Laplace,  and  S.  D.  Poisson.  To  exhibit  Laplace's  method  of 
determining  the  worth  of  candidates  by  combining  the  votes,  M.  W. 
Crofton  employs  the  fortuitous  division  of  a  straight  line.  This  in- 
volves, however,  an  a  priori  distribution  of  values  covering  evenly 
the  whole  range  from  o  to  loo.  Experience  shows  that  the  normal 
law  of  error  exhibits  a  more  correct  distribution.  On  this  point  Karl 
Pearson  produced  a  most  important  research.^  He  took  a  random 
sample  of  n  individuals  from  a  population  of  N  members  and  derived 
an  expression  for  the  average  difference  in  character  between  the  ^th 
and  the  {p+i)th  individual  when  the  sample  is  arranged  in  order  of 
magnitude  of  the  character.  H.  L.  Moore  of  Columbia  University 
has  attempted  to  trace  Pearson's  theory  in  the  statistics  relating  to 
the  efficiency  of  wages  (Economic  Journal,  Dec,  1907). 

Early  statistical  study  was  carried  on  under  the  name  of  "political 
arithinetic"  by  such  writers  as  Captain  John  Graunt  of  London  (1662) 
and  J.  P.  Siissmilch,  a  Prussian  clergyman  (17SS).  Application  of  the 
theory  of  probability  to  statistics  was  made  by  Edmund  Halley, 
Jakob  Bernoulli,  A.  De  Moivre,  L.  Euler,  P.  S.  Laplace,  and  S.  D. 
Poisson.  The  establishment  of  official  statistical  societies  and  statis- 
tical offices  was  largely  due  to  the  influence  of  the  Belgian  astronomer 
and  statistician,  Adolphe  Quetelet  (1796-1874)  of  the  obser\^atory 
at  Brussels,  "the  founder  of  modern  statistics."  Quetelet's  "average 
man"  in  whom  "all  processes  correspond  to  the  average  results 
obtained  for  society,"  who  "could  be  considered  as  a  type  of  the 
beautiful,"  has  given  rise  to  much  critical  discussion  by  Harold 
Westergaard  (1S90),  J,  Bertillon  (1S96),  A.  de  Fo\'ille  in  lus  "homo 
medius"  of  1907,  Joseph  Jacobs  in  his  "the  Middle  American" 
and  "the  Mean  Englishman."^  Quetelet's  visit  in  England  led 
to  the  organization,  in  1833,  of  the  statistical  section  of  the  British 
Association  for  the  Advancement  of  Science,  and  in  1835  of  the  Statis- 
tical Society  of  London.  Soon  after,  in  1S39,  was  formed  the  Amer- 
ican Statistical  Society.  Quetelet's  best  researches  on  the  application 
of  probability  to  the  physical  and  social  sciences  are  given  in  a  series 
of  letters  to  the  duke  of  Saxe-Coburg  and  Gotha,  Lettres  sur  la  theorie 
des  probabilites,  Brussels,  1846.  He  laid  emphasis  on  the  "law  of 
large  numbers,"  which  was  advanced  also  by  the  Frenchman  S.  D. 
Poisson  and  discussed  by  the  German  W.  Lexis  (1877),  the  Scandi- 
navians H.  Westergaard  and  Carl  Charlier,  and  the  Russian  Pafnuti 
Liwowich  Chebichev  (i 821-1894)  of  the  University  of  Petrograd  To 
Chebichev  we  owe  also  an  interesting  problem:  A  proper  fraction  being 

1  "Note  on  Francis  Gallon's  Problem,"  Biometrica,  Vol.  I,  pp.  390-390. 

2  See  Franz  Zizek's  Statistical  Averages,  transl.  by  W.  M.  Persons,  New  York. 
■•Oil:  D»  3'4 


ANALYSIS  381 

chosen  at  random,  what  is  the  probability  that  it  is  in  its  lowest 
terms? 

Of  the  different  kinds  of  averages,  A.  De  Morgan  concluded  that 
the  arithmetic  mean  represents  a  priori  the  most  probable  values. 
J.  W.  L.  Glaisher  took  exception  to  this.  G.  T.  Fcchner  investigated 
the  cases  where  the  "median"  (which  has  the  central  position  in  a 
series  of  items  arranged  according  to  size)  may  be  used  profitably. 
The  "mode,"  an  average  introduced  by  K.  Pearson  in  1895,  and 
used  by  0.  Udny  Yule,  has  been  applied  in  Germany  and  Austria 
to  the  fixing  of  workingmen's  insurance.  The  theory  of  averages  has 
been  studied  by  the  aid  of  the  calculus  of  probabilities  by  W.  Lexis, 
F.  Y.  Edgeworth  of  Oxford,  H.  Westergaard,  L.  von  Bortkewich,  G. 
T.  Fechner,  J.  von  Kries,  E.  Czuber  of  Vienna,  E.  Blaschke,  F.  Galton, 
K.  Pearson,  G.  U.  Y^ule,  and  A.  L.  Bowley  of  London.  A  few 
writers  take  the  ground  that  it  is  not  only  unnecessary  to  employ  prob- 
ability in  founding  statistical  theory,  but  that  it  is  inadvisable  to  do  so. 
Among  such  wTitcrs  are  G.  F.  Knapp  and  A.  AL  Guerry.^  The  Russian 
actuar}^  Jastremski  in  191 2  applied  the  Lexian  dispersion  theory  to  the 
testing  of  the  influence  of  medical  selection  in  life  insurance.  Other 
recent  publications  of  note  are  by  Lexis'  pupil  L.  von  Bortkewich  and 
by  Harold  Westergaard  of  Copenhagen.  Early  theories  of  popula- 
tion were  involved  in  much  confusion.  E.  Halley  and  some  eighteenth 
century  Avriters  proceeded  en  the  assumption  of  a  stationary  popu- 
lation. L.  Euler  adopted  the  hypothesis  that  the  yearly  births 
progress  in  a  geometric  series.  This  was  combatted  in  1839  by  L. 
Aloser,  while  G.  F.  Knapp  in  1868  represented  the  number  of  births 
and  deaths  as  a  continuous  function  of  the  time  and  of  the  age,  re- 
spectively. He  made  use  of  grapliic  representation.  G.  Zeuner  in 
1869  introduced  additional  geometric  and  analytic  aids.  In  1874 
Knapp  made  still  further  modifications,  allowing  for  discontinuous 
changes,  such  as  were  studied  also  by  W.  Lexis,  in  his  Theorie  der 
Bevdlkerungsstatistik,  Strassburg,  1875.  Formal  theories  of  popula- 
tion and  the  determination  oJ  mortality  were  investigated  also  by 
K.  Becker  in  1867  and  1874,  and  by  Th.  Wittstcin,  about  1881.  In 
1877  W.  Lexis  introduced  the  idea  of  "  dispersion  "and  "normal  dis- 
persion." Wilhelm  Lexis  (1837-1914)  became  in  1872  professor  at 
Strassburg,  in  1SS4  at  Breslau  and  in  18S7  at  Gottingen.  In  1893  he 
was  drawn  into  the  service  for  the  German  government. 

The  application  of  statistical  method  to  biology  was  begun  by 
Sir  Francis  Galton  (i'^22-i9ii),  "a  born  statistician."  Important 
is  his  Natural  Inherl'ancc,  1889,  in  which  he  uses  the  method  of  per- 
centiles, with  the  quartile  deviation  as  the  measure  of  dispersion.^ 
Two  other  Englishmen  entered  this  field  of  research,  Karl  Pearson 
of  University  College  and  W.  F.  R.  Weldon.     Pearson  developed 

^  Encyklop'adie  d.  Math.  Wissensck,  I  D  4a,  p.  822. 

^G.  Udny  Yule,  Theory  of  SialisUcs,  2.  Ed.,  London,  1912,  p.  154. 


382  A  HISTORY  OF  MATHEMATICS 

general  and  adequate  mathematical  methods  for  the  analysis  of 
biological  statistics.  To  him  are  due  the  terms  "mode,"  "standard 
deviation"  and  "coefficient  of  variation."  Before  him  the  "normal 
curve"  of  errors  had  been  used  exclusively  to  describe  the  distribution 
of  chance  events.  This  curve  is  symmetrical,  but  natural  phenomena 
sometimes  indicate  an  asymmetrical  distribution.  Accordingly 
Pearson,  in  his  Contributions  to  the  Theory  of  Evolution,  1899,  de- 
veloped skew  frequency  curves.  About  1890  the  Georg  Mendel  law 
of  inheritance  became  generally  known  and  caused  some  modification 
in  the  application  of  statistics  to  heredity.  Such  a  readjustment  was 
effected  by  the  Danish  botanist  W.  Johannsen.^ 

The  first  study  of  the  most  advantageous  combinations  of  data 
of  observation  is  due  to  Roger  Cotes,  in  the  appendix  to  his  Harmonia 
mensurarum,  1722,  where  he  assigns  weights  to  the  observations. 
The  use  of  the  arithmetic  mean  was  advocated  by  Thomas  Simpson 
in  a  paper  "An  attempt  to  show  the  advantage  arising  by  taking  the 
mean  of  a  number  of  observations,  in  practical  astronomy,"  ^  also  by 
J.  Lagrange  in  1773  and  by  Daniel  Bernoulli  in  1778.  The  first 
printed  statement  of  the  principle  of  least  squares  was  made  in  1806 
by  A.  M.  Legendre,  without  demonstration.  K.  F.  Gauss  had  used 
it  still  earlier,  but  did  not  publish  it  until  1809.  The  first  deduction 
of  the  law  of  probability  of  error  that  appeared  in  print  was  given  in 
1808  by  Robert  Adrain  in  the  Analyst,  a  journal  published  by  himself 
in  Philadelphia.  Of  the  earlier  proofs  given  of  this  law,  perhaps  the 
most  satisfactory  is  that  of  P.  S.  Laplace.  K.  F.  Gauss  gave  two 
proofs.  The  first  rests  upon  the  assumption  that  the  arithmetic 
mean  of  the  observations  is  the  most  probable  value.  Attempts  to 
prove  this  assumption  have  been  made  by  Laplace,  J.  F.  Encke  (1831), 
A.  De  Morgan  (1864),  G.  V.  Schiaparelli,  E.  J.  Stone  (1873),  and  A. 
Ferrero  (1876) .  Valid  criticisms  upon  some  of  these  investigations  were 
passed  by  J.  W.  L.  Glaisher.^  The  founding  of  the  Gaussian  proba- 
bility law  upon  the  nature  of  the  observed  errors  was  attempted  by 
F.  W.  Bessel  (1838),  G.  H.  L.  Hagen  (1837),  J.  F.  Encke  (1853),  P.  G. 
Tait  (1867),  and  M.  W.  Crofton  (1870).  That  the  arithmetic  mean, 
taken  as  the  most  probable  value,  is  not  under  all  circmnstances 
compatible  with  the  Gaussian  probabilty  law  has  been  shown  by 
Joseph  Bertrand  in  his  Calcul  des  probabilites  (1889),  and  by  others.^ 
The  development  of  the  theory  of  least  squares  along  practical  lines 
is  due  mainly  to  K.  F.  Gauss,  J.  F.  Encke,  P.  A.  Hansen,  Th.  Gallo- 
way, J.  Bienayme,  J.  Bertrand,  A.  Ferrero,  P.  Pizzetti.^     Simon 

1  Quart.  Pub.  Am.  Stat.  Ass'n,  N.  S.,  Vol.  XIV,  1914,  p.  45. 

^Miscellaneous  Tracts,  London,  1757. 

^  Land.  Astr.  Soc.  Mem.  39,  1872,  p.  75;  for  further  references,  see  Cyhlopadie  d. 
Math.  Wiss.,  I  D2,  p.  772. 

^  Consult  E.  L.  Dodd,  "  Probability  of  the  Arithmetic  Mean,  etc.,"  Annals  oj 
Mathematics,  2.  S.,  Vol.  14,  1913,  p.  186. 

^  E.  Czuber,  op.  cit.,  p.  179. 


ANALYSIS  383 

Newcomb  of  Washington  advanced  a  "generalized  theory  of  the 
comhinaLion  of  ol)servations  so  as  to  obtain  the  best  result,"  ^  when 
large  errors  arise  more  frequently  than  is  allowed  by  the  Gaussian 
probability  law.  The  same  subject  was  treated  by  R.  Lehmann- 
Filhes  in  Astrono?nische  Nachrichtcn,  1887. 

A  criterion  for  the  rejection  of  doubtful  observations  ^  was  given 
by  Benjamin  Peirce  of  Harvard.  It  was  accepted  by  the  American 
astronomers  B.  A.  Gould  (1824-1S96),  W.  Chauvenet  (1820-1870), 
and  J.  Winlock  (1826-1S75),  but  was  criticised  by  the  English  as- 
tronomer G.  B.  Airy.  The  prevailing  feeling  has  been  that  there 
exists  no  theoretical  basis  upon  which  such  criterion  can  be  rightly 
established. 

The  application  of  probability  to  epidemiology  was  first  considered 
by  Daniel  Bernoulli  and  has  more  recently  commanded  the  atten- 
tion of  the  English  statisticians  William  Farr  (1807-1S83),  John 
Brownlee,  Karl  Pearson,  and  Sir  Ronald  Ross.  Pearson  studied 
normal  and  abnormal  frequency  curves.  Such  curves  have  been 
fitted  to  epidemics  by  J.  BroAvnlee  in  1906,  S.  M.  Greenwood  in  igii 
and  1913,  and  Sir  Ronald  Ross  in  1916.^ 

Some  interest  attaches  to  the  discussion  of  whist  from  the  stand- 
point of  the  theory  of  probability,  as  is  contained  in  William  Pole's 
Philosophy  of  Whist,  New  York  and  London,  1883.  The  problem 
is  a  generalization  of  the  game  of  "treize"  or  "recontre,"  treated 
by  Pierre  R.  de  Montmort  in  170S. 

Differential  Equations.    Difference  Equations 

Criteria  for  distinguishing  between  singular  solutions  and  particular 
solutions  of  dilYcrential  equations  of  the  first  order  were  advanced 
by  A.  ]M.  Legendre,  S.  D.  Poisson,  S.  F.  Lacroix,  A.  L.  Cauchy,  and 
G.  Boole.  After  J.  Lagrange,  the  c-discrimJnant  relation  commanded 
the  attention  of  Jean  Marie  Constant  Duhamel  (1797-1872)  of  Paris, 
C.  L.  M.  H.  Navier,  and  others.  But  the  entire  theory  of  singular 
solutions  was  re-investigated  about  1870  along  new  paths  by  J.  G. 
Darboux,  A.  Cayley,  E.  C.  Catalan,  F.  Casorati,  and  others.  The 
geometric  side  of  the  subject  was  considered  more  minutely  and  the 
cases  were  explained  in  which  Lagrange's  method  does  not  yield 
singular  solutions.  Even  these  researches  were  not  altogether  satis- 
factory as  they  did  not  furnish  necessary  and  sufficient  conditions 
for  singular  solutions  which  depend  on  the  differential  equation  alone 
and  not  in  an)'^vay  upon  the  general  solution.  Returning  to  more 
purely  analytical  considerations  and  building  on  work  of  Ch.  Briot 
and  J.  C.  Bouquet  of  1856,  Carl  Schmidt  of  Giessen  in  1884,  H.  B. 
Fine  of  Princeton  in  1890,  and  Meyer  Llamburger  (1S3S-1903)  of 

1  Am.  Jour.  Math.,  Vol.  8,  1886,  p.  343. 

■"-Gould  Astr.  Jour.,  II,  185.'. 

■^Nature,  Vol.  97,  191(1,  p.  243. 


384  A  HISTORY  OF  MATHEMATICS 

Berlin  brought  the  problem  to  final  solutions.     Active  in  this  line 
were  also  John  MuUer  Hill  and  A.  R.  Forsyth.^ 

The  first  scientific  treatment  of  partial  differential  equations  was 
given  by  J.  Lagrange  and  P.  S.  Laplace.  These  equations  were  in- 
vestigated in  more  recent  time  by  G.  Monge,  J.  F.  Pfaff,  C.  G.  J. 
Jacobi,  Emile  Bour  (1S31-1S66)  of  Paris,  A.  Weiler,  R.  F.  A.  Clebsch, 
A.  N.  Korkine  of  St.  Petersburg,  G.  Boole,  A.  Meyer,  A.  L.  Cauchy, 
J.  A.  Serret,  Sophus  Lie,  and  others.  In  1873  their  reseaches,  on 
partial  differential  equations  of  the  first  order,  were  presented  i:i 
text-book  form  by  Paul  Mansion,  of  the  University  of  Gand.  Pro- 
ceeding to  the  consideration  of  some  detail,  we  remark  that  the  keen 
researches  of  Johann  Friedrich  Ffaff  (1765-1825)  marked  a  decided 
advance.  He  was  an  intimate  friend  of  K.  F.  Gauss  at  Gottingen. 
Afterwards  he  was  with  the  astronomer  J.  E.  Bode.  Later  he  became 
professor  at  Helmstadt,  then  at  Halle.  By  a  peculiar  method,  Pfaff 
found  the  general  integration  of  partial  differential  equations  of  the 
first  order  for  any  number  of  variables.  Starting  from  the  theory  of 
ordinary  differential  equations  of  the  first  order  in  n  variables,  he 
gives  first  their  general  integration,  and  then  considers  the  integra- 
tion of  the  partial  differentiai  equations  as  a  particular  case  of  the 
former,  assuming,  however,  as  known,  the  general  integration  of 
differential  equations  of  any  order  between  two  variables.  His  re- 
searches led  C.  G.  J.  Jacobi  to  introduce  the  name  "Pfaffian  prob- 
lem." From  the  connection,  observed  by  W.  R.  Hamilton,  between 
a  system  of  ordinary  differential  equations  (in  analytical  mechanics) 
and  a  partial  differential  equation,  C.  G.  J.  Jacobi  drew  the  conclu- 
sion that,  of  the  series  of  systems  whose  successive  integration  Pfaff 's 
method  demanded,  all  but  the  first  system  were  entirely  superfluous. 
R.  F.  A.  Clebsch  considered  Pfaff's  problem  from  a  new  point  of  view, 
and  reduced  it  to  systems  of  simultaneous  linear  partial  differential 
equations,  which  can  be  established  independently  of  each  other  with- 
out any  integration.  Jacobi  materially  advanced  the  theory  of  dif- 
ferential equations  of  the  first  order.  The  problem  to  determine  un- 
known functions  in  such  a  way  that  an  integral  containing  these  func- 
tions and  their  differential  coefficients,  in  a  prescribed  manner,  shall 
reach  a  maximum  or  minimum  value,  demands,  in  the  first  place, 
the  vanishing  of  the  first  variation  of  the  integral.  This  condition 
leads  to  differential  equations,  the  integration  of  which  determines  the 
functions.  To  ascertain  whether  the  value  is  a  maximum  or  a  mini- 
mum, the  second  variation  must  be  examined.  This  leads  to  new  and 
difficult  differential  equations,  the  integration  of  which,  for  the  simpler 
cases,  was  ingeniously  deduced  by  C.  G.  J.  Jacobi  from  the  integra- 
tion of  the  differential  equations  of  the  first  variation.  Jacobi's 
solution  was  perfected  by  L.  O.  Hesse,  while  R.  F.  A.  Clebsch  extended 

'  We  have  used  S.  Rothenbers;,  "Geschichte  .  .  .  der  singularen  Losungen"  in 
Abh.  z.  Gesch.  d.  Malh.  Wissensch.  (]\I.  Cantor),  Heft  XX,  3.  Leipzig,  1908. 


ANALYSIS  385 

to  the  general  case  Jacobi's  results  on  the  second  variation.  A.  L. 
Cauchy  gave  a  method  of  solving  partial  differential  equations  of  the 
Arst  order  having  any  number  of  variables,  which  was  corrected  and 
extended  by  J.  A.  Serret,  J.  Eertrand,  O.  Bonnet  in  France,  and 
\Vassili  Grigorjcwich  Imshenetski  (1832-1892)  of  the  University  of 
Charkow  in  Russia.  Fundamental  is  the  proposition  of  Cauchy  that 
e\'ery  ordinary  differential  equation  admits  in  the  vicinity  of  any  non- 
singular  point  of  an  integral,  which  is  synectic  within  a  certain  circle 
of  convergence,  and  is  developable  by  Taylor's  theorem.  Allied  to  the 
point  of  view  indicated  by  this  theorem  is  that  of  G.  F.  B.  Riemann, 
who  regards  a  function  of  a  single  variable  as  defined  by  the  position 
and  nature  of  its  singularities,  and  who  has  applied  this  conception 
to  that  linear  differential  equation  of  the  second  order,  which  is  satis- 
fied by  the  hypergeomctric  series.  This  equation  was  studied  also 
by  K.  F.  Gauss  and  E.  E.  Kummer.  Its  general  theory,  when  no  re- 
striction is  imposed  upon  the  value  of  the  variable,  has  been  considered 
by  J.  Tannery,  of  Paris,  who  employed  L.  Fuchs'  method  of  hnear 
differential  equations  and  found  all  of  Kummer's  twenty-four  inte- 
grals of  this  equation.  This  study  has  been  continued  by  Edouard 
Goursat  (1858-  ),  professor  of  mathematical  analysis  in  the  Uni- 
versity of  Paris.  His  activities  have  been  in  the  theory  of  functions, 
pseudo-  and  hyper-elliptic  integrals,  differential  equations,  invariants 
and  surfaces.  Jules  Tannery  (1848-1910)  became  professor  of  me- 
chanics at  the  Sorbonne  in  1875,  and  sub-director  at  the  Ecole  Normale 
in  Paris  in  1884.  His  researches  have  been  in  the  field  of  analysis  and 
the  theory  of  functions. 

As  outlined  by  A.  R.  Forsyth  ^  in  iqoS,  the  status  of  partial  differen- 
tial equations  is  briefly  as  follows:  .'■Jnce  the  posthumous  publication, 
in  1862,  of  C.  G.  J.  Jacobi's  treatment  of  partial  differential  equations 
of  the  first  order  involving  only  one  dependent  variable,  or  a  system 
of  such  equations,  it  may  be  said  that  we  have  a  complete  method 
of  formal  integration  of  such  equations.  In  the  formal  integration  of  a 
partial  differential  equation  of  the  second  or  higher  orders  new  dif- 
ficulties are  encountered.  Only  in  rare  instances  is  direct  integration 
possible.  The  known  normal  types  of  integrals  even  for  such  equa- 
tions of  only  the  second  order  are  few  in  number.  The  primitive  may 
be  given  by  means  of  a  single  relation  between  the  variables,  or  by 
means  of  a  number  of  equations  involving  eliminable  parameters  (such 
as  the  customary  forms,  due  to  A.  M.  Legendre,  G.  Monge,  or  K. 
Weierstrass,  of  the  primitive  of  minimal  surfaces),  or  by  means  of  a 
relation  involving  definite  integrals  arising  in  problems  in  physics. 
"With  all  these  types  of  primitives,"  says  A.  R.  Forsyth,^  "it  being 
assumed  that  immediate  and  direct  integration  is  impossible — ,  a 

'  A.  R.  Forsyth  in  Alii,  del  IV.  Congr.  Intern.,  Roma,  1008.  Vol.  I,  Roma,  1.909, 
p.  90. 

-A.  R.  Forsyth,  loc.  cit.,  p.  90.    We  are  summarizing  part  of  this  article. 


386  A  HISTORY  OF  MATHEMATICS 

primitive  is  obtained  by  the  use  of  processes,  that  sometimes  are  frag- 
mentary in  theory,  usually  are  tentative  in  practice  and  nearly  always 
are  indirect  in  the  sense  that  they  are  compounded  of  a  number  of 
formal  operations  having  no  organic  relation  with  the  primitive. 
In  such  circumstances  ...  is  the  primitive  completely  comprehen- 
sive of  all  the  integrals  belonging  to  the  equation?"  A.  M.  Ampere  in 
1815  propounded  a  broad  definition  of  a  general  integral — one  in 
which  the  only  relations,  which  subsist  among  the  variables  and  the 
derivatives  of  the  dependent  variable  and  which  are  free  from  the 
arbitrary  elements  in  the  integral,  are  constituted  by  the  differential 
equation  itself  and  by  equations  deduced  from  it  by  differentiation. 
This  definition  is  incomplete  on  various  grounds.  E.  Goursat  gave  in 
1898  a  simple  instance  to  show  that  an  integral  satisfying  all  of  Am- 
pere's requirements  was  not  general.  A  second  definition  of  a  general 
integral  was  given  in  1S89  by  J.  G.  Darboux,  based  on  A.  L.  Cauchy's 
existence-theorem:  An  integral  is  general  when  the  arbitrary  ele- 
ments which  it  contains  can  be  specialized  in  such  a  way  as  to  provide 
the  integral  established  in  that  theorem.  This  definiiion,  according 
to  A.  R.  Forsyth,  calls  for  a  more  careful  discussion  of  obvious  and 
latent  singularities. 

There  are  three  principal  methods  of  proceeding  to  the  construc- 
tion of  an  integral  of  partial  differential  equations  of  the  second  order, 
which  lead  to  success  in  special  cases.  One  method  given  by  P.  S. 
Laplace  in  1777  applies  to  linear  equations  with  two  independent 
variables.  It  can  be  used  for  equations  of  order  higher  than  the 
second.  It  has  been  developed  by  J.  G.  Darboux  and  V.  G.  Imshenet- 
ski,  1872.  A  second  method,  originated  by  A.  M.  Ampere,  while 
general  in  spirit  and  in  form,  depends  upon  individual  skill  unassisted 
by  critical  tests.  Later  researches  along  this  line  are  due  to  E.  Borel 
(1895)  and  E.  T.  Whittaker  (1903).  A  third  method  is  due  to  J.  G. 
Darboux  and  includes,  according  to  A.  R.  Forsyth's  classification, 
the  earlier  work  of  ]\Ionge  and  G.  Boole.  As  first  given  by  J.  G.  Dar- 
boux in  1870,  it  applied  only  to  the  case  of  two  independent  variables, 
but  it  has  been  extended  to  equations  of  more  than  two  independent 
variables  and  orders  higher  than  the  second;  it  is  not  universally 
effective.  "Such  then,"  says  Forsyth,  "are  the  principal  methods 
hitherto  devised  for  the  formal  Integra *^.ion  of  partial  equations  of  the 
second  order.  They  have  been  discussed  by  many  mathematicians 
and  they  have  been  subjected  to  frec-ient  modifications  in  details: 
but  the  substance  of  the  processes  remains  unaltered." 

Instances  are  known  in  ordinary  linear  equations  when  the  primitives 
can  be  expressed  by  definite  integrals  or  by  means  of  asymptotic 
expansions,  the  theory  of  which  owes  much  to  H.  Poincare.  Such 
instances  within  the  region  of  partial  equations  are  due  to  E.  Borel. 

G.  F.  B.  Riemann  had  remarked  in  1857  that  functions  expressed 
by  K.  F.  Gauss'  hyi^ergeometric  series  F  (a,  6,  y,  x),  wliich  satisfy 


ANALYSIS  387 

a  homogeneous  linear  differential  equation  of  the  second  order  with 
rational  coefficients,  might  be  utilized  in  the  solution  of  any  linear  dif- 
ferential equation.^  Another  mode  of  solving  such  equations  was  due 
to  Cauchy  and  was  extended  by  C.  A.  A.  Briot  and  J.  C.  Bouquet)  and 
consisted  in  the  development  into  power  series.  The  fertility  of  the 
conceptions  of  G.  F.  B.  Riemann  and  A.  L.  Cauchy  with  regard  to 
differential  equations  is  attested  by  the  researches  to  which  they  have 
given  rise  on  the  part  of  Lazarus  Fuchs  (1833-1902)  of  Berlin.  Fuchs 
was  born  in  Moschin,  near  Posen,  and  became  professor  at  the  Uni- 
versity of  Berlin  in  18S4.  In  1S65  L.  Fuchs  combined  the  two  methods 
in  the  study  of  linear  differential  equations:  One  method  using  power- 
series,  as  elaborated  by  A.  L.  Cauchy,  C.  A.  A.  Briot,  and  J.  C.  Bou- 
quet; the  other  method  using  the  hypergeometric  series  as  had  been 
done  by  G.  F.  B.  Riemann.  By  this  union  Fuchs  initiated  a  new 
theory  of  linear  differential  equations.^  Cauchy's  development  into 
power-series  together  with  the  calcul  des  limites,  afforded  existence 
theorems  which  are  essentially  the  same  in  nature  as  those  relating  to 
differential  equations  in  general.  The  singular  points  of  the  linear 
differential  equation  received  attention  also  from  G.  Frobenius  in 
1874,  G.  Peano  in  1889,  M.  Bocher  in  1901.  A  second  approach  to 
existence  theorems  was  by  successive  approximation,  first  used  in 
1864  by  J.  Caque,  then  by  L.  Fuchs  in  1870,  and  later  by  H.  Poin- 
care  and  G.  Peano.  A  third  line,  by  interpolation,  is  originally  due  to 
A.  L.  Cauchy  and  received  special  attention  from  V.  Volterra  in  1887. 
The  general  theory  of  linear  differential  equations  received  the  atten- 
tion of  L.  Fuchs,  and  of  a  large  number  of  workers,  including  C.  Jordan, 
V.  Volterra,  and  L.  Schlesinger.  Singular  places  where  the  solutions 
are  not  indeterminate  were  investigated  by  J.  Tannery,  L.  Schlesinger, 
G.  J.  Wallenberg,  and  many  others.  Ludwig  Wilhelm  Thome  (1841- 
1910)  of  the  University  of  Greifswald,  discovered  in  1877  what  he 
called  normal  integrals.  Divergent  series  which  formally  satisfy  dif- 
ferential equations,  first  noticed  by  C.  A.  A.  Briot  and  J.  C.  Bouquet 
in  1856,  were  first  seriously  considered  by  H.  Poincare  in  1885  who 
pointed  out  that  such  series  may  represent  certain  solutions  asymp- 
totically. Asymptotic  representations  have  been  examined  by  A. 
Kneser  (1S96),  E.  Picard  (1S96),  J.  Horn  (1897),  and  A.  Hamburger 
(1905).  A  special  type  of  linear  differential  equation,  the  "Fuchsian 
type,"  with  coefficients  that  are  single- valued  (eindeutig),  and  the 
solutions  of  which  have  no  points  of  indeterminateness,  was  investi- 
gated by  Fuchs,  and  it  was  found  that  the  coefficients  of  such  an  equa- 
tion are  rational  functions  of  x.  Studies  based  on  analogies  of  linear 
differential  equations  with  algebraic  equations,  first  undertaken  by 

^  We  are  using  L.  Schlesinger,  Entwickelung  d.  Thcorie  d.  linearen  Differattial- 
gleichtmgen  seit  1865,  Leipzig  and  Berlin,  1909. 

2  We  are  using  here  a  report  by  L.  Schlesinger  in  Jahresh.  d.  d.  Math.  Vereinigung, 
Vol.  18,  1909,  pp.  133-260. 


388  \  HISTORY  OF  MATHEMATICS 

N.  H.  Abel,  J.  Liouville  and  C.  G.  J.  Jacobi,  were  pursued  later  by  P. 
Appell  (1880),  by  E.  Picard  who  worked  under  the  influence  of  S. 
Lie's  theory  of  transformation  groups,  and  by  an  army  of  workers  in 
France,  England,  Germany,  and  the  United  States.  The  consideration 
of  differential  invariants  enters  here.  Lame's  differential  equation, 
considered  by  him  in  1S57,  was  taken  up  by  Ch.  Hermite  in  1877  and 
soon  after  in  still  more  generalized  form  by  L.  Fuchs,  F.  Brioschi, 
E.  Picard,  G.  M.  Mittag-Leffler  and  F.  Klein. 

The  analogies  of  linear  differential  equations  with  algebraic  func- 
tions, problems  of  inversion  and  uniformization,  as  well  as  questions 
involving  group  theory  received  the  attention  of  the  analysts  of  the 
second  half  of  the  century. 

The  theory  of  invariants  associated  with  linear  differential  equations 
as  developed  by  Halphen  and  by  A.  R.  Forsyth  is  closely  connected 
with  the  theory  of  functions  and  of  groups.  Endeavors  have  thus 
been  made  to  determine  the  nature  of  the  function  defined  by  a  dif- 
ferential equation  from  the  differential  equation  itself,  and  not  from 
any  analytical  expression  of  the  function,  obtained  first  by  solving 
the  differential  equation.  Instead  of  studying  the  properties  of  the 
integrals  of  a  differential  equation  for  all  the  values  of  the  variable, 
investigators  at  first  contented  themselves  with  the  study  of  the  prop- 
erties in  the  vicinity  of  a  given  point.  The  nature  of  the  integrals 
at  singular  points  and  at  ordinary  points  is  entirely  different.  Charles 
Augiiste  Albert  Briot  (1817-1882)  und  Jean  Claude  Bouquet  (1819-1885) 
both  of  Paris,  studied  the  case  when,  near  a  singular  point,  the  dif- 
ferential equations  take  the  form  (x— .to)-t-=  I   (xy).    L.  Fuchs  gave 

the  development  in  series  of  the  integrals  for  the  particular  case  of 
linear  equations.  H.  Poincare  did  the  same  for  the  case  when  the 
equations  are  not  linear,  as  also  for  partial  differential  equations  of 
the  first  order.  The  developments  for  ordinary  points  were  given 
by  A.  L.  Cauchy  and  Sophie  Kovalevski  (1850-1891).  Madame 
Kovalevski  was  born  at  Moscow,  was  a  pupil  of  K.  Weierstrass  and 
became  professor  of  Analysis  at  Stockholm. 

Henri  Poincare  (1854-1912)  was  born  at  Nancy  and  commenced 
his  studies  at  the  Lycee  there.  While  taking  high  rank  as  a  student, 
he  did  not  display  exceptional  precocity.  He  attended  the  Ecole 
Polytechnique  and  the  Ecole  Nationale  Super  ieure  des  Mines  in  Paris, 
receiving  his  doctorate  from  the  University  of  Paris  in  1879.  He  be- 
came instructor  in  mathematical  analysis  at  the  University  of  Caen. 
In  1 88 1  he  occupied  the  chair  of  physical  and  experimental  mechanics 
at  the  Sorbonne,  later  the  chair  of  mathematical  physics  and,  after 
the  death  of  F.  Tisserand,  the  chair  of  mathematical  astronomy  and 
celestial  mechanics.  Although  he  did  not  reach  old  age,  he  published 
numerous  books  and  more  than   1500  memoirs.     Probably  neither 


ANALYSIS  389 

A.  L.  Cauchy  nor  even  L.  Euler  equalled  him  in  the  quantity  of  scien- 
tific productions.  P.  Painleve  said  that  everyone  of  his  many  papers 
carried  the  mark  of  a  lion.  Poincare  wrote  on  mathematics,  physics, 
astrononiy,  and  pliilosophy.  No  other  scientist  of  his  day  was  able  to 
work  in  such  a  wide  range  of  subjects.  INIany  consider  him  the  great- 
est mathematician  of  his  time.  Each  year  he  lectured  on  a  ditlerent 
subject;  these  lectures  were  reported  and  jjublished  by  his  former 
students.  In  this  manner  were  brought  out  works  on  capillarity, 
elasticity,  Newtonian  potential,  vortices,  the  propagation  of  heat, 
thermodynamics,  light,  electric  oscillations,  electricity  and  optics, 
Hertzian  oscillations,  mathematical  electricity,  kinematics,  equili- 
brium of  fluid  masses,  celestial  mechanics,  general  astronomy,  prob- 
ability. His  popular  works  on  the  philosophy  of  science.  La  science  et 
I'liypotliese  (1902),  La  valeiir  de  la  science  (1905),  Science  et  methode 
(190S)  have  been  translated  into  German,  in  part  also  into  Spanish, 
Hungarian,  and  Japanese.  An  English  translation  by  George  Bioicc 
Halsted  appeared  in  one  volume  in  1913. 

Our  numerous  references  to  Poincare  will  indicate  that  he  wrote 
on  nearly  every  branch  of  pure  mathematics.  Says  F.  R.  Moulton:  * 
''The  importance  of  his  papers  can  be  inferred  from  the  enormous 
number  of  references  to  his  theorems  in  all  modem  treatises,  espe- 
cially on  the  various  branches  of  analysis.  The  emphasis  on  analysis 
dojs  not  mean  that  he  neglected  geometry,  analysis  situs,  groups, 
numbjr  theory,  or  the  foundations  of  mathematics,  for  he  illuminated 
all  these  subjects  and  others;  but  it  is  placed  there  because  this  do- 
main includes  his  researches  on  differential  equations,  dating  from  his 
doctor's  dissertation  to  very  recent  times,  his  contributions  to  the 
theory  of  functions,  and  his  discovery  of  fuchsian  and  theta-fuchsian 
functions.  His  command  of  the  powerful  methods  of  modern  analysis 
was  positively  dazzling."  As  to  his  method  of  work  E.  Borel  says: 
"The  method  of  Poincare  is  essentially  active  and  constructive.  He 
approaches  a  question,  acquaints  himself  with  its  present  condition 
without  being  much  concerned  about  its  history,  finds  out  immediately 
the  new  analytical  formulas  by  which  the  question  can  be  advanced, 
deduces  hastily  the  essential  results,  and  then  passes  to  another  ques- 
tion. After  having  finished  the  wTiting  of  a  memoir,  he  is  sure  to 
pause  for  a  while,  and  to  think  out  how  the  exposition  could  be  im- 
proved; but  he  would  not,  for  a  single  instance,  indulge  in  the  idea  of 
devoting  several  days  to  didactic  work.  Those  days  could  be  better 
utilized  in  exploring  new  regions."  Poincare  tells  how  he  came  to 
make  his  first  mathematical  discoveries:  "For  a  fortnight  I  labored 
to  demonstrate  that  there  could  exist  no  function  analogous  to  those 
that  I  have  since  called  the  fuchsian  fimctions.    I  was  then  very  ig- 

^  Popular  Astronomy,  Vol.  20,  1912.  We  are  usins;  also  Ernest  Lebon,  Ilenri 
Poincare,  Biographic,  Paris,  1909;  "Jules  Henri  Poincare"  in  Nature,  Vol.  90, 
London,  1912,  p.  353;  George  Sarton,  "Henri  Poincare''  in  Ciel  et  Terre,  1913^ 


390  A  HISTORY  OF  MATHEMATICS 

TiOrant.  Every  day  1  seated  myself  at  my  work  table  and  spent  an 
hour  or  two  there,  tryhig  a  great  many  combinations,  but  I  arrived  at 
no  result.  One  night  when,  contrary  to  my  custom,  I  had  taken  black 
coffee  and  I  could  not  sleep,  ideas  surged  up  in  crowds.  I  felt  them  as 
they  struck  against  one  another  until  two  of  them  stuck  together,  so 
to  speak,  to  form  a  stable  combination.  By  morning  I  had  established 
the  existence  of  a  class  of  fuchsian  functions,  those  which  are  derived 
from  the  hypergeometric  series.  I  had  merely  to  put  the  results  in 
shape,  which  only  took  a  few  hours."  ^ 

Poincare  enriched  the  theory  of  integrals.  The  attempt  to  express 
integrals  by  developments  that  are  always  convergent  and  not  limited 
to  particular  points  in  a  plane  necessitates  the  introduction  of  new 
transcendents,  for  the  old  functions  permit  the  integration  of  only  a 
s'^all  number  of  differential  equations.  H.  Poincare  tried  this  plan 
With  linear  equations,  which  were  then  the  best  known,  having  been 
studied  in  the  vicinity  of  given  points  by  L.  Fuchs,  L.  W.  Thome,  G. 
Frobenius,  H.  A.  Schwarz,  F.  Klein,  and  G.  H.  Halphen.  Confining 
himself  to  those  with  rational  algebraical  coefificients,  H.  Poincare  was 
able  to  integrate  them  by  the  use  of  functions  named  by  him  Fiich- 
sians?  He  divided  these  equations  into  "families."  If  the  integral 
of  such  an  equation  be  subjected  to  a  certain  transformation,  the 
result  will  be  the  integral  of  an  equation  belonging  to  the  same  family. 
The  new  transcendents  have  a  great  analogy  to  elliptic  functions; 
while  the  region  of  the  latter  may  be  divided  into  parallelograms,  each 
representing  a  group,  the  former  may  be  divided  into  curvilinear 
polygons,  so  that  the  knowledge  of  the  function  inside  of  one  polygon 
carries  with  it  the  knowledge  of  it  inside  the  others.  Thus  H.  Poin- 
care arrives  at  what  he  calls  Fuchsian  groups.  He  found,  moreover, 
that  Fuchsian  functions  can  be  expressed  as  the  ratio  of  two  trans- 
cendents (theta-fuchsians)  in  the  same  way  that  elliptic  functions  can 
be.  If,  instead  of  linear  substitutions  with  real  coefficients,  as  em- 
ployed in  the  above  groups,  imaginary  coefficients  be  used,  then  dis- 
continuous  groups  are  obtained,  which  he  called  Kleinians.  The  ex- 
tension to  non-linear  equations  of  the  method  thus  applied  to  linear 
equations  was  begun  by  L.  Fuchs  and  H.  Poincare. 

Much  interest  attaches  to  the  determination  of  those  linear  differ- 
ential equations  which  can  be  integrated  by  simpler  functions,  such 
as  algebraic,  elliptic,  or  Abelian.  This  has  been  studied  by  C.  Jordan, 
P.  Appell  of  Paris,  and  H.  Poincare. 

Paul  Appell  (1855-  )  was  born  in  Strassburg.  After  the  an- 
nexation of  Alsace  to  Germany  in  1871,  he  emigrated  to  Nancy  to 
escape  German  citizenship.     Later  he  studied  in  Paris  and  in  1886 

1  H.  Poincare,  The  Foundations  of  Sciaicf,  transl.  by  G.  B.  Halsted,  The  Science 
Press,  New  York  and  Garrison,  N.  Y.,  1913,  p.  387. 

2  Henri  Poincare,  Notice  sur  les  Travaux  Scientifiques  de  Henri  PoincarS,  Paris, 
1886,  p.  9. 


ANALYSIS  391 

became  professor  of  mechanics  there.  His  researches  are  in  analysis, 
function  theory,  infinitesimal  geometry  and  rational  mechanics. 

Whether  an  ordinary  differential  equation  has  one  or  more  solutions 
which  satisfy  certain  terminal  or  boundary  conditions,  and,  if  so,  what 
the  character  of  these  solutions  is,  has  received  renewed  attention 
the  last  quarter  century  by  the  consideration  of  finer  and  more  remote 
questions.^  Existence  theorems,  oscillation  properties,  asymptotic 
expression:.,  development  theorems  have  been  studied  by  David  Hil- 
bert  of  Gottingen,  Maxime  Bocher  of  Harvard,  Max  Mason  of  the 
University  of  Wisconsin,  Alauro  Picone  of  Turin,  R.  M.  E.  JVIises  of 
Strassburg,  H.  Weyl  of  Gottingen  and  especially  by  George  D.  Birk- 
hotT  of  Harvard.  Integral  equations  have  been  used  to  some  extent 
in  ooundary  problems  of  one  dimension;  "this  method  would  seem, 
however,  to  be  chietly  valuable  in  the  cases  of  two  or  more  dimensions 
where  many  of  the  simplest  questions  are  still  to  be  treated." 

A  standard  text-book  on  Dijfereniial  Equations,  including  original 
matter  on  integrating  factors,  singular  solutions,  and  especially  on 
symbolical  methods,  was  prepared  in  1859  by  G.  Boole. 

A  Treatise  on  Linear  Differential  Equations  (1889)  was  brought  out 
by  Thomas  Craig  of  the  Johns  Hopkins  University.  He  chose  the 
algebraic  method  of  presentation  followed  by  Ch.  Hermite  and  H. 
Poincare,  instead  of  the  geometric  method  preferred  by  F.  Klein  and 
H.  A.  Schwarz.  A  notable  work,  the  Traite  d'Analyse,  1891-1896, 
was  published  by  Emile  Picard  of  Paris,  the  interest  of  which  was  made 
to  centre  in  the  subject  of  differential  equations.  A  second  edition 
has  appeared. 

Simple  difference  equations  or  "finite  differences"  were  studied  by 
eighteenth  century  mathematicians.  When  in  1882  H.  Poincare  de- 
veloped the  novel  notion  of  asymptotic  representation,  he  applied  it 
to  linear  difference  equations.  In  recent  years  a  new  type  of  problem 
has  arisen  in  connection  with  them.  It  looks  now  as  if  the  continuity 
of  nature,  which  has  been  for  so  long  assumed  to  exist,  were  a  fiction 
and  as  if  discontinuities  represented  the  realities.  "It  seems  almost 
certain  that  electricity  is  done  up  in  pellets,  to  which  we  have  given 
the  name  of  electrons.  That  heat  comes  in  quanta  also  seems  prob- 
able." ^  Much  of  theory  based  on  the  assumption  of  continuity  may 
be  found  to  be  mere  approximation.  Homogeneous  linear  difference 
equations,  not  intimately  bound  up  with  continuity,  were  taken  up  in- 
dependently by  investigators  widely  apart.  In  1909  Niels  Erik  Nor- 
lund  of  the  Univ^ersity  of  Lund  in  Sweden,  Henri  Galbrun  of  I'Ecole 
Normale  in  Paris  and,  in  191 1,  R.  D.  Carmichael  of  the  University  of 
Illinois  entered  this  field  of  research,  Carmichael  used  a  method  of 
successive  approximation  and  an  extension  of  a  contour  integral  due  to 

'  See  a  historical  summary  by  Maxime  B6cher  in  Proceed,  of  the  jln  intern.  Con- 
gress, Cambridge,  1912,  Vol.  I,  Cambridp^e,  1913,  p.  163. 
2  R.  D.  Carmichael  in  Science,  N.  S.  Vol.  45,  19 17,  p.  472. 


392  A  HISTORY  OF  MATHEMATICS 

C.  Guichard.  G.  D.  Birkhoff  of  Harvard  made  important  contribu- 
tions showing  the  existence  of  certain  intermediate  solutions  and  of  the 
principal  solutions.  The  asymptotic  form  of  these  solutions  is  de- 
termined by  him  throughout  the  complex  plane.  The  extension  to 
non-homogeneous  equations  of  results  reached  for  homogeneous  ones 
has  been  made  by  K.  P.  Williams  of  the  University  of  Indiana.^ 

Integral  Equations,  Integra-differential  Equations,  General  Analysis, 
Functional  Calculus 

The  mathematical  perplexities  which  led  to  the  invention  of  integral 
equations  were  stated  by  J.  Hadamard  ^  in  191 1  as  follows:  "Those 
problems  (such  as  Dirichlet's)  exercised  the  sagacity  of  geometricians 
and  were  the  object  of  a  great  deal  of  important  and  well-known  work 
through  the  whole  of  the  nineteenth  century.  The  very  variety  of 
ingenious  methods  applied  showed  that  the  question  did  not  cease  to 
preserve  its  rather  mysterious  character.  Only  in  the  last  years  of 
the  century  were  we  able  to  treat  it  with  some  clearness  and  under- 
stand its  true  nature.  .  .  .  Let  us  therefore  inquire  by  what  device 
this  new  view  of  Dirichlet's  problem  was  obtained.  Its  peculiar  and 
most  remarkable  feature  consists  in  the  fact  that  the  partial  dififerential 
equation  is  put  aside  and  replaced  by  a  new  sort  of  equation,  namely, 
the  integral  equation.  This  new  method  makes  the  matter  as  clear 
as  it  was  formerly  obscure.  In  many  circumstances  in  modern  analysis, 
contrary  to  the  usual  point  of  view,  the  operation  of  integration  proves 
a  much  simpler  one  than  the  operation  of  derivation.  An  example  of 
this  is  given  by  integral  equations  where  the  unknown  function  is 
written  under  such  signs  of  integration  and  not  of  differentiation.  The 
type  of  equation  which  is  thus  obtained  is  much  easier  to  treat  than 
the  partial  differential  equation.  The  type  of  integral  equations 
corresponding  to  the  plane  Dirichlet  problem  is 

(i)  4>{x) -  X  J VOO  K  (x,y)  dy=f{x), 

where  0  is  the  unknown  function  of  x  in  the  interval  (A,  B),/and  K 
are  known  functions,  and  X  is  a  known  parameter.  The  equations 
of  the  elliptic  type  in  many-dimensional  space  give  similar  integral 
equations,  containing  however  multiple  integrals  and  several  inde- 
pendent variables.  Before  the  introduction  of  equations  of  the  above 
type,  each  step  in  the  study  of  elliptic  partial  differential  equations 
seemed  to  bring  with  it  new  difficulties.  .  .  .  [But]  an  equation  such 
as  (i)  .  .  .  gives  all  the  required  results  at  once  and  for  all  the  pos- 
sible types  of  such  problems.  .  .  .    Previously,  in  the  calculation  of 

^  Trans.  Am.  Math.  Soc,  Vol.  14,  1913,  p.  209. 

^  J.  Hadamard,  Four  Lectures  on  Mathematics  delivered  at  Columbia  University  in 
igii,  New  York,  1915,  pp.  12-15. 


ANALYSIS  3Q3 

the  resonance  of  a  room  filled  with  air,  the  shape  of  the  resonator 
had  to  be  quite  simple,  which  requirement  is  not  a  necessary  one  for 
the  case  where  integral  equations  are  employed.  We  need  only  make 
the  elementary  calculation  of  the  function  K  and  apply  to  the  function 
so  calculated  the  general  method  of  resolution  of  integral  equations." 

The  new  departure  in  analysis  was  made  in  1900  by  Eric  Ivar 
Fredholm  (1866-  ),  a  native  of  Stockholm,  who  in  1898  was  docent 
at  the  University  of  Stockholm  and  later  became  connected  with  the 
imperial  bureau  of  insurance.  In  a  paper,^ ''  Sur  une  nouvelle  methode 
pour  la  resolution  du  probleme  de  Dirichlet,"  1900,  he  studied  an 
integral  equation  from  the  point  of  view  of  an  immediate  generaliza- 
tion of  a  system  of  linear  equations.  Integral  equations  bear  the 
same  relation  to  the  integral  calculus  as  difTeren:ial  equations  do  to 
the  differential  calculus.  Before  this  time  certain  integral  equations 
had  received  the  attention  of  N.  H.  Abel,  J.  Liouville,  and  Eugene 
Rouche  (1S32-1910)  of  Paris,  but  were  quite  neglected. 

Abel  had  in  1S23  proposed  a  generalization  of  the  tautochrone 
problem,  the  solution  of  which  involved  an  integral  equation  that 
has  since  been  designated  as  of  the  first  kind.  Liouville  in  1837 
showed  that  a  particular  solution  of  a  linear  differential  equation  of 
the  second  order  could  be  found  by  solving  an  integral  equation, 
now  designated  as  of  the  second  kind.  A  method  of  solving  integral 
equations  of  the  second  kind  was  given  by  C.  Neumann  (1S87).  The 
term  "integral  equation"  is  due  to  P.  du  Bois-Reymond  (Crelle, 
Vol.  103,  1SS8,  p.  22S)  who  exemplified  the  danger  of  making  pre- 
dictions by  the  declaration  that  "the  treatment  of  such  equations 
seems  to  present  insuperable  difficulties  to  the  analysis  of  to-day." 
The  recent  theory  of  integral  equations  owes  its  origin  to  specific 
problems  in  mechanics  and  mathematical  physics.  Since  1900  these 
equations  have  been  used  in  the  study  of  existence  theorems  in  the 
theory  of  potential;  they  were  employed  in  1904  by  W.  A.  Stekloff 
and  D.  Hilbert  in  the  consideration  of  boundary  values  and  in  matters 
relating  to  Fourier  series,  by  Henri  Poincare  in  the  study  of  tides  and 
Hertzian  waves.  Linear  integral  equations  present  many  analogies 
with  linear  algebraic  equations.  While  E.  I.  Fredholm  used  the  theory 
of  algebraic  equations  merely  to  suggest  theorems  on  integral  equa- 
tions, which  were  proved  independently,  D.  Hilbert  in  his  early  work 
on  this  subject  followed  the  process  of  taking  limits  in  the  results  of 
algebraic  theory.  Hilbert  has  introduced  the  term  "kernel"  of  linear 
integral  equations  of  the  first  and  second  kind.  The  theory  of  integral 
equations  has  been  advanced  by  Erhard  Schmidt  of  Breslau  and 
Vito  Volterra  of  Rome.  Systematic  treatises  on  integral  equations 
have  been  prepared  by  Maxime  Bocher  of  Harvard  (1909),  Gerhard 
Kowalewski  of  Prag  (1900),  Adolf  Kneser  of  Breslau  (191 1),  T. 
Lalesco  of  Paris  (1912),  H.  B.  Heywood,  and  Al.  Frechet  (1912). 
^bjversigl  af  akadcmicns  JJrhandlingar  57,  Stockholm,  1900. 


394  A  HISTORY  OF  MATHEMATICS 

Maxime  Bocher  (1867-1918)  was  born  in  Boston  and  graduated 
at  Harvard  in  1888.  After  three  years  of  study  at  Gottingen  he  re- 
turned to  Harvard  where  he  was  successively  instructor,  assistant- 
professor  and  professor  of  mathematics.  He  was  president  of  the 
American  Mathematical  Society  in  1909-1910.  Among  his  works  are 
ReihenenMkkelungen  der  Potential-theorie,  1891,  enlarged  in  1894,  and 
Leqons  sur  les  Melhodes  de  Sturm,  containing  the  author's  lectures  de- 
livered at  the  Sorbonne  in  1913-1914. 

A.  Voss  in  1 913  stresses  the  value  of  integral  equations  thus:  ^ 
"In  the  last  ten  years  ...  the  theory  of  integral  equations  has 
attained  extraordinary  importance,  because  through  them  problems 
in  the  theory  of  differential  equations  may  be  solved  which  previously 
could  be  disposed  of  only  in  special  cases.  We  abstain  from  sketching 
their  theory,  which  makes  use  of  infinite  determinants  that  belong 
to  linear  equations  with  an  infinite  number  of  unknowns,  of  quadratic 
forms  with  infinitely  many  variables,  and  which  has  succeeded  in 
throwing  new  light  upon  the  great  problems  of  pure  and  applied 
mathematics,  especially  of  mathematical  physics." 

Important  advances  along  the  line  of  a  "general  analysis"  and  its 
application  to  a  generalization  of  the  theory  of  linear  integral  equa- 
tions have  been  made  since  1906  by  E.  H.  Moore  of  the  University 
of  Chicago.^  From  the  existence  of  analogies  in  different  theories  he 
infers  the  existence  of  a  general  theory  comprising  the  analogous 
theories  as  special  cases.  He  proceeds  to  a  "unification,"  resulting, 
first,  from  the  recent  generalization  of  the  concept  of  independent 
variable  effected  by  passing  from  the  consideration  of  variables  defined 
for  all  points  in  a  given  interval  to  that  of  variables  defined  for  all 
points  in  any  given  set  of  points  lying  in  the  range  of  the  variable, 
secondly,  from  the  consideration  of  functions  of  an  infinite  as  well 
as  a  finite  number  of  variables,  and,  thirdly,  from  a  still  further  gen- 
eralization which  leads  him  to  functions  of  a  "general  variable." 
E.  H.  Moore's  general  theory  includes  as  special  cases  the  theories  of 
E.  I.  Fredholm,  D.  Hilbert,  and  E.  Schmidt.  G.  D.  Birkhoff  in  191 1 
presented  the  following  birds'-eye  view  of  recent  movements:  ^  "Since 
the  researches  of  G.  W.  Hill,  V.  Volterra,  and  E.  I.  Fredholm  in  the 
direction  of  extended  linear  systems  of  equations,  mathematics  has 
been  in  the  way  of  great  development.  That  attitude  of  mind  which 
conceives  of  the  function  as  a  generaHzed  point,  of  the  method  of 
successive  approximation  as  a  Taylor's  expansion  in  a  function  va- 
riable, of  the  calculus  of  variations  as  a  limiting  form  of  the  ordinary 
algebraic  problem  of  maxima  and  minima  is  now  crystallizing  into  a 
new  branch  of  mathematics  under  the  leadership  of  S.  Pincherle,  J. 

^  A.  Voss,  Ueber  das  Wesen  d.  Math.,  1913,  p.  63. 

*See  Proceed.  5th  Intern,  Congress  of  Mathematicians,  Cambridge,  19 13,  Vo^  I, 
p.  230. 
»  Bull.  Am.  Mjth.  Soc,  Vol.  17,  1911,  p.  415. 


ANALYSIS  395 

Hadamard,  D.  Hilbert,  E.  H.  Moore,  and  others.  For  this  field 
Professor  Moore  proposes  the  term  'General  Analysis,'  defined  as 
'the  theory  of  systems  of  classes  of  functions,  functional  operations, 
etc.,  involving  at  least  one  general  variable  on  a  general  range.'  He 
has  fixed  attention  on  the  most  abstract  aspect  of  this  field  by  con- 
sidering functions  of  an  absolutely  general  variable.  The  nearest 
approach  to  a  similar  investigation  is  due  to  M.  Frechet  (Paris  thesis, 
1906),  who  restricts  himself  to  variables  for  which  the  notion  of  a 
Hmiting  value  is  valid."  Researches  along  the  line  of  E.  H.  Moore's 
"General  Analysis"  arc  due  to  A.  D.  Pitcher  of  Adelbert  College  and 
E.  W.  Chittenden  of  the  University  of  Illinois.  In  his  "General 
Analysis"  Moore  defines  "complete  independence"  of  postulates 
which  has  received  the  further  attention  of  E.  V.  Huntington,  R.  D. 
Beetle,  L.  L.  Dines,  and  M.  G.  Gaba. 

V.  Volterra  discusses  integro-diflferential  equations  which  involve 
not  only  the  unknown  functions  under  signs  of  integration  but  also 
the  unkno\\'n  functions  themselves  and  their  derivatives,  and  shows 
their  use  in  mathematical  physics.  G.  C.  Evans  of  the  Rice  Institute 
extended  A.  L.  Cauchy's  existence  theorem  for  partial^  differential 
equations  to  integro-differential  equations  of  the  "static  type"  m 
which  the  variables  of  differentiation  are  different  from  those  of  in- 
tegration. Mixed  linear  integral  equations  have  been  discussed  by 
W.  A.  Hurwatz  of  Cornell  University. 

The  study  of  integral  equations  and  the  theory  of  point  sets  has  led 
to  the  development  of  a  body  of  theory  called  functional  calculus. 
One  part  of  this  is  the  theory  of  the  functions  of  a  line.  As  early  as 
1887  Vito  Volterra  of  the  University  of  Rome  developed  the  funda- 
mental theory  of  what  he  called  functions  depending  on  other  func- 
tions and  functions  of  curves.  Any  quantity  which  depends  for  its 
value  on  the  arc  of  a  curve  as  a  whole  is  called  a  function  of  the  fine. 
The  relationships  of  functions  depending  on  other  functions  are 
called  "  f onctionelles "  by  J.  Hadamard  in  his  Leqons  sur  le  calcul  des 
variations,  1910,  and  "functional"  by  English  writers.  Functional 
equations  and  systems  of  functional  equations  have  received  the  atten- 
tion of  Grifhth  C.  Evans  of  the  Rice  Institute,  Luigi  Smigallia  of 
Pavia,  Giovanni  L.  T.  C.  Giorgi  of  Rome,  A.  R.  Schweitzer  of  Chicago, 
Eric  H.  Neville  of  Cambridge,  and  others.  Neville  solves  the  race- 
course puzzle  of  covering  a  circle  by  a  set  of  five  circular  discs.  Says 
G.  B.  Mathews:  ^  "We  must  express  our  regret  that  English  math- 
ematics is  so  predominantly  analytical.  Cannot  some  one,  for  in- 
stance, give  us  a  truly  geometrical  theory  of  J.  V.  Poncelet's  poristic 
polygons,  or  of  von  Staudt's  thread-constructions  for  conicoids?  "  In 
the  theory  of  functional  equations,  "a  single  equation  or  a  system  of 
equations  expressing  some  property  is  taken  as  the  definition  of  a 
class  of  functions  whose  characteristics,  particular  as  well  as  collective, 
^Nature,  Vol.  gy,  1916,  p.  398. 


396  A  HISTORY  OF  MATHEMATICS 

are  to  be  developed  as  an  outcome  of  the  equations"  (E.  B.  Van 
Vleck). 

An  important  generalization  of  Fourier  series  has  been  made, 
"and  we  have  a  great  class  of  expansions  in  the  so-called  orthogonal 
and  biorthogonal  functions  arising  in  the  study  of  differential  and 
integral  equations.  In  the  field  of  differential  equations  the  most 
important  class  of  these  functions  was  first  defined  in  a  general  and 
explicit  manner  (in  1907)  by  .  .  .  G.  D.  Birkhoff  of  Harvard  Univer- 
sity; and  their  leading  fundamental  properties  were  developed  by 
him."  ^  In  boundary  value  problems  of  differential  equations  which 
are  not  self-adjoint,  biorthogonal  systems  of  functions  play  the  same 
role  as  the  orthogonal  systems  do  in  the  self-adjoint  case.  Anna  J. 
Pell  established  theorems  for  biorthogonal  systems  analogous  to  those 
of  F.  Riesz  and  E.  Fischer  for  orthogonal  systems. 

Theories  of  Irrationals  and  Theory  of  Aggregates 

The  new  non-metrical  theories  of  the  irrational  were  called  forth 
by  the  demands  for  greater  rigor.  The  use  of  the  word  "quantity" 
as  a  geometrical  magnitude  without  reference  to  number  and  also 
as  a  number  which  measures  some  magnitude  was  disconcerting,  es- 
pecially as  there  existed  no  safe  ground  for  the  assumption  that  the 
same  rules  of  operation  applied  to  both.  The  metrical  view  of  number 
involved  the  entire  theory  of  measurement  which  assumed  greater 
difficulties  with  the  advent  of  the  non-Euclidean  geometries.  In  at- 
tempts to  construct  arithmetical  theories  of  number,  irrational  num- 
bers were  a  source  of  trouble.  It  was  not  satisfactory  to  operate  with 
irrational  numbers  as  if  they  were  rational.  What  are  irrational 
numbers?  Considerable  attention  was  paid  to  the  definition  of  them 
as  limits  of  certain  sequences  of  rational  numbers.  A.  L.  Cauchy  in 
his  Cmirs  d' Analyse,  182 1,  p.  4,  says  "an  irrational  number  is  the 
limit  of  diverse  fractions  which  furnish  more  and  more  approximate 
v^alues  of  it."  Probably  Cauchy  was  satisfied  of  the  existence  of 
irrationals  on  geometric  grounds.  If  not,  his  exposition  was  a  rea- 
soning in  a  circle.  To  make  this  plain,  suppose  we  have  a  develop- 
ment of  rational  numbers  and  we  desire  to  define  limit  and  also  irra- 
tional number.  With  Cauchy  we  may  say  that  "when  the  successive 
values  attributed  to  a  variable  approach  a  fixed  value  indefinitely 
so  as  to  end  by  differing  from  it  as  little  as  is  wished,  this  fixed  value 
is  called  the  limit  of  all  the  others."  Since  we  are  still  confined  to 
the  field  of  rational  numbers,  this  limit,  if  not  rational,  is  non-existent 
and  fictitious.  If  now  we  endeavor  to  define  irrational  number  as  a 
limit,  we  encounter  a  break-down  in  our  logical  development.  It 
became  desirable  to  define  irrational  number  arithmetically  without 
reference  to  limits.  Thi?  was  achieved  independently  and  at  almost 
1  R.  D.  Carmichael  in  Science,  N.  S.,  Vol.  45,  1917,  p.  471. 


ANALYSIS 


397 


Ihe  same  time  by  four  men,  Charles  Meray  (1835-191 1),  K.  Weier- 
strass,  R.  Dedekind,  and  Georg  Cantor.  Meray 's  first  publication  was 
in  the  Revue  des  societes  savantes:  sc.  math.  (2)  4,  1869,  p.  284;  his  later 
publications  were  in  1872,  1887,  1894.  Meray  was  bom  at  Chalons 
in  France  and  was  professor  at  the  University  of  Dijon.  The  earliest 
publication  of  K.  Weierstrass's  presentation  was  made  by  H.  Kossak 
in  1872.  In  the  same  year  it  was  published  in  Crelle's  Journal,  Vol.  74, 
p.  174,  by  E.  Heine  who  had  received  it  from  Weierstrass  by  oral 
communication.  R.  Dedekind's  publication  is  Stetigkeit  und  irra- 
tionalc  Zahlen,  Braunschweig,  1872.  In  1888  appeared  his  Was  sind 
und  was  sollen  die  Zahlen?  Richard  Dedekind  (1831-1916)  was  bom 
in  Braunschweig,  studied  at  Gottingen  and  in  1854  became  privat- 
docent  there.  From  185S  to  1862  he  was  professor  at  the  Polytech- 
nicum  in  Zurich  as  the  successor  of  J.  L.  Raabe,  and  from  1863  to 
1894  professor  at  the  Technical  High  School  in  Braunschweig.  He 
worked  on  the  theory  of  numbers.  The  substance  of  his  Stetigkeit  und 
Irrationale  Zahlen,  was  worked  out  by  him  before  he  left  Zurich.  He 
worked  also  in  function  theory.  Georg  Cantor's  first  printed  statement 
was  in  Mathem.  Annalen,  Vol.  5,  1872.  p.  123.^  Georg  Cantor  (1845- 
1918)  was  bom  at  Petrograd,  lived  from  1856  to  1863  in  South  Ger- 
many, studied  from  1S63  to  1869  at  Berlin  where  he  came  under  the 
influence  of  Weierstrass.  While  in  Berlin  he  once  defended  the  re- 
markable thesis:  In  mathematics  the  art  of  properly  stating  a  question 
is  more  important  than  the  solving  of  it  (In  re  mathematica  ars  pro- 
ponendi  quasstionem  pluris  facienda  est  quam  solvendi).  He  became 
privatdocent  at  Halle  in  1869,  extraordinary  professor  in  1872  and 
ordinary  professor  in  1879.  In  recent  years  he  has  suffered  from  ill 
health,  taking  the  form  of  mental  disturbances.  When  emerging 
from  such  attacks,  his  mind  is  said  to  be  most  productive  of  scientific 
results.  Nearly  all  his  papers  are  on  the  development  of  the  theory 
of  aggregates.  It  had  been  planned  to  hold  on  March  3,  1915,  an 
international  celebration  of  his  seventieth  birthday,  but  on  account 
of  the  war,  only  a  few  German  friends  gathered  at  Halle  to  do  him 
honor. 

In  the  theories  of  Ch.  Meray  and  G.  Cantor  the  irrational  number 
is  obtained  by  an  endless  sequence  of  rational  numbers  a\,  02, 03,  .  .  . 
which  have  the  property  \a„~am\  <€,  provided  n  and  m  are  sufficiently 
great.  The  method  of  K.  Weierstrass  is  a  special  case  of  this.  R. 
Dedekind  defined  every  "cut"  in  the  system  of  rational  numbers  to 
be  a  number,  the  "open  cuts"  constituting  the  irrational  numbers. 
To  G.  Cantor  and  Dedekind  we  owe  the  important  theory  of  the  linear 
continuum  which  represents  the  culmination  of  efforts  which  go  back 
to  the  church  fathers  of  the  Middle  Ages  and  the  writings  of  Aristotle. 
By  this  modern  continuum,  "the  notion  of  number,  integral  or  frac- 

1  For  details  see  Encyclopedic  des  sciences  mathematiques,  Tome  I,  Vol.  I,  1904: 
§§6-8,  pp.  147-15^- 


398  A  HISTORY  OF  MATHEMATICS 

tional,  has  been  placed  upon  a  basis  entirely  independent  of  meas^ar- 
able  magnitude,  and  pure  analysis  is  regarded  as  a  scheme  which  deals 
with  number  only,  and  has,  per  se,  no  concern  with  measurable  quan- 
tity. Analysis  thus  placed  upon  an  arithmetical  basis  is  characterized 
by  the  rejection  of  all  appeals  to  our  special  intuitions  of  space,  time 
and  motion,  in  support  of  the  possibility  of  its  operations"  (E.  W. 
Hobson).  The  arithmetization  of  mathematics,  which  was  in  progress 
during  the  entire  nineteenth  century,  but  mainly  during  the  time  of 
Ch.  Meray,  L.  Kronecker,  and  K.  Weierstrass,  was  characterized  by 
E.  W.  Hobson  in  1902  in  the  following  terms:  ^  ''In  some  of  the  text- 
books in  common  use  in  this  country,  the  symbol  00  is  still  used  as  if 
it  denoted  a  number,  and  one  in  all  respects  on  a  par  with  the  finite 
numbers.  The  foundations  of  the  integral  calculus  are  treated  as  if 
Riemann  had  never  lived  and  worked.  The  order  in  which  double 
limits  are  taken  is  treated  as  immaterial,  and  in  many  other  respects 
the  critical  results  of  the  last  century  are  ignored.  .  .  . 

"The  theory  of  exact  measurement  in  the  domain  of  the  ideal  ob- 
jects of  abstract  geometry  is  not  immediately  derivable  from  intuition, 
but  is  now  usually  regarded  as  requiring  for  its  development  a  previous 
independent  investigation  of  the  nature  and  relations  of  number. 
The  relations  of  number  having  been  developed  on  an  independent 
basis,  the  scheme  is  applied  by  the  help  of  the  principle  of  congruency, 
or  other  equivalent  principle,  to  the  representation  of  extensive  or 
intensive  magnitude.  .  .  .  Tliis  complete  separation  of  the  notion 
of  number,  especially  fractional  number,  from  that  of  magnitude, 
involves,  no  doubt,  a  reversal  of  the  historical  and  psychological 
orders.  .  .  .  The  extreme  arithmetizing  school,  of  which,  perhaps, 
L.  Kronecker  was  the  founder,  ascribes  reality,  whatever  that  may 
mean,  to  integral  numbers  only,  and  regards  fractional  numbers  as 
possessing  only  a  derivative  character,  and  as  being  introduced  only 
for  convenience  of  notation.  The  ideal  of  this  school  is  that  every 
theorem  of  analysis  should  be  interpretable  as  giving  a  relation  be- 
tween integral  numbers  only.  .  .  . 

"The  true  ground  of  the  difficulties  of  the  older  analysis  as  regards 
the  existence  of  limits,  and  in  relation  to  the  application  to  measur- 
able quantity,  lies  in  its  inadequate  conception  of  the  domain  of 
number,  in  accordance  with  which  the  only  numbers  really  defined 
were  rational  jiumbers.  This  inadequacy  has  now  been  removed  by 
means  of  a  purely  arithmetical  definition  of  irrational  numbers,  by 
means  of  which  the  continuum  of  real  numbers  has  been  set  up  as 
the  domain  of  the  independent  variable  in  ordinary  analysis.  This 
definition  has  been  given  in  the  main  in  three  forms — one  by  E.  Heine 
and  G.  Cantor,  the  second  by  R.  Dedekind,  and  the  third  by  K.  Weier- 
strass. Of  these  the  first  two  are  the  simplest  for  working  purposes, 
and  are  essentially  equivalent  to  one  another;  the  difference  between 
^Proceed.  London  Math.  Soc,  Vol.  35,  1902,  pp.  117-139;  see  p.  118. 


ANALYSIS  399 

ihem  is  that,  while  Dcdekind  defines  an  irrational  number  by  means 
af  a  section  of  all  the  rational  numbers,  in  the  Heine-Cantor  form  of 
definition  a  selected  convergent  aggregate  of  such  numbers  is  em- 
ployed. The  essential  change  introduced  by  this  definition  of  irra- 
tional numbers  is  that,  for  the  scheme  of  rational  numbers,  a  new 
scheme  of  numbers  is  substituted,  in  which  each  number,  rational  or 
irrational,  is  defined  and  can  be  exhibited  in  an  indefinitely  great 
number  of  ways,  by  means  of  a  convergent  aggregate  of  rational 
numbers.  ...  By  this  conception  of  the  domain  of  number  the  root 
difficulty  of  the  older  analysis  as  to  the  existence  of  a  limit  is  turned, 
each  number  of  the  continuum  being  really  defined  in  such  a  way  that 
it  itself  exhibits  the  limit  of  certain  classes  of  convergent  sequences.  .  .  . 
It  should  be  observed  that  the  criterion  for  the  convergence  of  an 
aggregate  is  of  such  a  character  that  no  use  is  made  in  it  of  infinitesi- 
mals, definite  finite  numbers  alone  being  used  in  the  tests.  The  old 
attempts  to  prove  the  existence  of  limits  of  convergent  aggregates 
were,  in  default  of  a  previous  arithmetical  definition  of  irrational 
number,  doomed  to  inevitable  failure.  ...  In  such  applications  of 
analysis — as,  for  example,  the  rectification  of  a  curve — the  length  of 
the  curve  is  defined  by  the  aggregate  formed  by  the  lengths  of  a  proper 
sequence  of  inscribed  polygons.  ...  In  case  the  aggregate  is  not 
convergent,  the  curve  is  regarded  as  not  rectifiable.  .  .  . 

"It  has  in  fact  been  showoi  that  many  of  the  properties  of  functions, 
such  as  continuity,  differentiability,  are  capable  of  precise  definition 
when  the  domain  of  the  variable  is  not  a  continuum,  provided,  how- 
ever, that  domain  is  perfect;  this  has  appeared  clearly  in  the  course 
of  recent  investigations  of  the  properties  of  non-dense  perfect  aggre- 
gates, and  of  functions  of  a  variable  whose  domain  is  such  an  aggre- 
gate." 

In  191 2  Philip  E.  B.  Jourdain  of  Fleet,  near  London,  characterized 
theories  of  the  irrational  substantially  as  follows:^  "Dedekind's 
theory  had  not  for  its  object  to  prove  the  existence  of  irrationals: 
it  showed  the  necessity,  as  Dedekind  thought,  for  the  mathematician 
to  create  them.  In  the  idea  of  the  creation  of  numbers,  Dedekind 
was  followed  by  O.  Stolz;  but  H.  Weber  and  M.  Pasch  showed  how 
the  supposition  of  this  creation  could  be  avoided:  H.  Weber  defined 
real  numbers  as  sections  {Schnitte)  in  the  series  of  rationals;  M.  Pasch 
(like  B.  Russell)  as  the  segments  which  generate  these  sections.  In 
K.  Weierstrass'  theory,  irrationals  were  defined  as  classes  of  rationals. 
Hence  B.  Russell's  objections  (stated  in  his  Principles  of  Mathematics, 
Cambridge,  1903,  p.  2S2)  do  not  hold  against  it,  nor  does  Russell 
seem  to  credit  Weierstrass  and  Cantor  with  the  avoidance  of  quite 
the  contradiction  that  they  did  avoid.  The  real  objection  to  Weier- 
strass' theory,  and  one  of  the  objections  to  G.  Cantor's  theory,  is 

'  P.  E.  B.  Jourdain,  "On  Isoid  Relations  and  Theories  of  Irrational  Number"  in 
Inlernaiional  Congre^is  of  Malhematicians,  Cambridge,  19 12. 


400  A  HISTORY  OF  MATHEMATICS 

that  equality  has  to  be  re-defined.  In  the  various  arithmetical  thcori  - 
of  irrational  numbers  there  are  three  tendencies:  (a)  the  number  is 
defined  as  a  logical  entity — a  class  or  an  operation — ,  as  with  K. 
Weierstrass,  H.  Weber,  M.  Pasch,  B.  Russell,  M.  Fieri;  (b)  it  is 
"created,"  or,  more  frankly,  postulated,  as  with  R.  Dedekind,  O. 
Stolz,  G.  Peano,  and  Ch.  Meray;  (c)  it  is  defined  as  a  sign  (for  what, 
is  left  indeterminate),  as  with  E.  Heine,  G.  Cantor,  H.  Thomae,  A, 
Pringsheim.  ...  In  the  geometrical  theories,  as  with  Paul  du  Bois- 
Reymond,  a  real  number  is  a  sign  for  a  length.  In  B.  Russell's  theory 
it  appears  to  be  equally  legitimate  to  define  a  real  number  in  various 
ways." 

The  theory  of  aggregates  (Mengenlehre,  theorie  des  ensembles, 
theory  of  sets)  owes  its  development  to  the  endeavor  to  clarify  the 
concepts  of  independent  variable  and  of  function.  Formerly  the 
notion  of  an  independent  variable  rested  on  the  na.'ve  concept  of  the 
geometric  continuum.  Now  the  independent  variable  is  restricted 
to  some  aggregate  of  values  or  points  selected  out  of  the  continuum. 
The  term  function  was  destined  to  receive  various  definitions.  J. 
Fourier  advanced  the  theorem  that  an  arbitrary  function  can  be 
represented  by  a  trigonometric  series.  P.  G.  L.  Dirichlet  looked  upon 
the  general  functional  concept  as  equivalent  to  any  arbitrary  table  of 
values.  When  G.  F.  B.  Riemann  gave  an  example  of  a  function  ex- 
pressed analytically  which  was  discontinuous  at  each  rational  point, 
the  need  of  a  more  comprehensive  theory  became  evident.  The  first 
attempts  to  meet  the  new  needs  were  made  by  Hermann  Hankel  and 
Paul  du  Bois-Reymond.  The  Allgemeine  Funktionentheorie  of  du 
Bois-Reymond  brilliantly  sets  forth  the  problems  in  philosophical 
form,  but  it  remained  for  Georg  Cantor  to  advance  and  develop  the 
necessary  ideas,  involving  a  treatment  of  infinite  aggregates.  Even 
though  the  infinite  had  been  the  subject  of  philosophic  contemplation 
for  more  than  two  thousand  years,  G.  Cantor  hesitated  for  ten  years 
before  placing  his  ideas  before  the  mathematical  public.  The  theory 
of  aggregates  sprang  into  being,  as  a  science,  when  G.  Cantor  intro- 
duced the  notion  of  "enumerable"  aggregates.^  G.  Cantor  began  his 
publications  in  1870;  in  1883  he  published  his  Grundlagen  einer  all- 
gemeinen  Mannichjaltigkeitslehre.  In  1895  and  1897  appeared  in 
M athematische  Annalen  his  Bcitrdge  zur  Begrwidung  der  transfiniten 
Mengenlehre}  These  researches  have  played  a  most  conspicuous  role 
not  only  in  the  march  of  mathematics  toward  logical  exactitude,  but 
also  in  the  realm  of  philosophy. 

G.  Cantor's  theory  of  the  continuum  was  used  by  P.  Tannery  in 
1885  in  the  search  for  a  profounder  view  of  Zeno's  arguments  against 

^  A.  Schoenflies,  Entwickelung  der  Mengenlehre  und  Hirer  Anwrndungen,  gemeinsam 
mil  Hans  Hahn  hcransgegeben,  Leipzig  u.  Berlin,  1913,  p.  2. 

2  Translated  into  English  by  Philip  E.  B.  Jourdain  and  published  by  the  Open 
Court  Puhl.  Co.,  Chicago,  1915. 


ANALYSIS  401 

motion.  Paul  Tannery  (1S43-IQ04),  a  brother  of  Jules  Tannery, 
attended  the  Ecole  Polytechnique  in  Paris  and  then  entered  the  state 
corps  of  manufacturing  engineers.  He  devoted  his  days  to  business 
and  his  evenings  to  the  study  of  the  history  of  science.  From  1802 
to  i8q6  he  held  the  chair  of  Greek  and  Latin  philosophy  at  the  College 
de  France,  but  later  he  failed  to  receive  the  appointment  to  the  chair 
of  the  history  of  the  sciences,  although  he  was  the  foremost  French 
historian  of  his  day.  He  was  a  deep  student  of  Greek  scientists,  par- 
ticularly of  Diophantus.  Other  historical  periods  were  taken  up  after- 
wards, 'particularlv  that  of  R.  Descartes  and  P.  Fermat.  His  re- 
searches, consisting  mostly  of  separate  articles,  are  being  republished 
in  collected  form. 

In  1883  G.  Cantor  stated  that  every  set  and  in  particular  the  con- 
tinuum can  be  well-ordered.  In  1904  and  1907  E.  Zermelo  gave 
proofs  of  this  theorem,  but  they  have  not  been  generally  accepted  and 
have  given  rise  to  much  discussion.  G.  Peano  objected  to  Zermelo's 
proof  because  it  rests  on  a  postulate  ("Zermelo's  principle")  expressing 
a  property  of  the  continuum.  In  1907  E.  Zermelo  formulated  that 
postulate  thus:  "A  set  S  which  is  divided  into  subsets  A,  B,  C,  .  .  ., 
each  containing  at  least  one  element,  but  containing  no  elements  in 
common,  contains  at  least  one  subset  S,  which  has  just  one  element  in 
common  with  each  of  the  subsets  A,  B,  C,  .  .  ."  {Math.  Ann.,  Vol. 
56,  p.  no).  To  this  G.  Peano  objects  that  one  may  not  apply  an 
infinite  number  of  times  an  arbitrary  law  by  virtue  of  which  one  cor- 
relates to  a  class  some  member  of  that  class.  E.  B.  Wilson  comments 
on  this:  "Here  are  two  postulates  by  two  different  authorities;  th^ 
postulates  are  contradictory,  and  each  thinker  is  at  liberty  to  adopt 
whichever  appears  to  him  the  more  convenient."  Zermelo's  postulate, 
before  it  had  been  formulated,  was  tacitly  assumed  in  the  researches 
of  R.  Dedekind,  G.  Cantor,  F.  Bernstein,  A.  Schonflies,  J.  Konig,  and 
others.  Zermelo's  proof  was  rejected  by  H.  Poincare,  E.  Borel,  R. 
Baire,  V.  A.  Lebesgue.^  A  third  proof  that  every  set  can  be  well- 
ordered  was  given  in  1915  by  Friedrich  M.  Hartogs  of  Munich.  P. 
E.  B.  Jourdain  of  Fleet  did  not  consider  this  proof  altogether  satis- 
factory. In  1918  he  gave  one  of  his  own  {Mind,  27,  386-388)_  and 
declared,  "  thus  any  aggregate  can  be  well  ordered,  Zermelo's  '  axiom ' 
can  be  proved  quite  generally,  and  Hartog's  work  is  completed." 

In  1897  Cesare  Burali-Forti  of  Turin  pointed  out  the  following  para- 
dox: The  series  of  ordinal  numbers,  which  is  well-ordered,  must  have 
the  greatest  of  all  ordinal  numbers  as  its  order  type.  Yet  the  type  of 
the  above  series  of  ordinal  numbers,  when  followed  by  its  t\T3e,  must 
be  a  greater  ordinal  number,  for  /3+i  is  greater  than  /3.  Therefore,  a 
well-ordered  series  of  ordinal  numbers  containing  all  ordinal  numbers 
itself  defines  a  new  ordinal  number  not  included  in  the  original  series. 

Another  paradox,  due  to  Jules  Antoine  Richard  of  Chateauroux 
I  See  Bulletin  Am.  Math.  Soc,  Vol.  14,  p.  438- 


A02  A  HISTORY  OF  MATHEMATICS 

in  1906,  relates  to  the  aggregate  of  decimal  fractions  between  o  and  i 
which  can  be  defined  by  a  finite  number  of  words;  a  new  decimal  frac- 
tion can  be  defined,  which  is  not  included  in  the  previous  ones. 

Bertrand  Russell  discovered  another  paradox,  given  in  his  Prin- 
ciples of  Mathematics,  1903,  pp.  364-3G8,  101-107),  which  is  stated  by 
Philip  E.  B.  Jourdain  thus:  "If  w  is  the  class  of  all  those  terms  x  such 
that  X  is  not  a  member  of  x,  then,  if  w  is  a  member  of  w,  it  is  plain  that 
w  is  not  a  member  of  w;  while  if  w  is  not  a  member  of  w,  it  is  equally 
plain  that  ■zt'  is  a  member  of  zc."  ^  These  paradoxes  are  closely  allied 
to  the  "Epimenides  puzzle":  Epimenides  was  a  Cretan  who  said  that 
all  Cretans  were  liars.  Hence,  if  his  statement  was  true  he  was  a  liar. 
H.  Poincare  and  B.  Russell  attribute  the  paradoxes  to  the  open  and 
clandestine  use  of  the  word  "all."  The  difficulty  lies  in  the  definition 
of  the  word  "Menge." 

Noteworthy  among  the  attempts  to  place  the  theory  of  aggregates 
upon  a  foundation  tJiat  will  exclude  the  paradoxes  and  antinomies  that 
had  arisen,  was  the  formulation  in  1907  of  seven  restricting  axioms 
by  E.  Zermelo  in  Math.  Annalen,  65,  p.  261. 

Julius  Konig  (1849-1913),  the  Hungarian  mathematician,  in  his 
Neue  Grundlagen  der  Logik,  Arithmetik  urui  Mengenlehre,  1914,  speaks 
of  E.  Zermelo's  axiom  of  selection  (Auswahlaxiom)  as  being  really 
a  logical  assumption,  not  an  axiom  in  the  old  sense,  whose  freedom 
from  contradiction  must  be  demonstrated  along  with  the  other 
axioms.  He  takes  pains  to  steer  clear  of  the  antinomies  of  B.  Russell 
and  C.  Burali-Forti.  For  a  discussion  of  the  logical  and  philosophical 
questions  involved  in  the  theory  of  aggregates,  consult  the  second 
edition  of  E.  Borel's  Leqons  sur  la  theorie  des  fonclions,  Paris,  1914, 
note  IV,  which  gives  letters  written  by  J.  Hadamard,  E.  Borel,  H. 
Lebesgue,  R.  Baire,  touching  the  validity  of  Zermelo's  demonstration 
that  the  linear  continuum  is  well-ordered.  A  set  of  axioms  of  ordinal 
magnitude  was  given  by  A.  B,  Frizell  in  191 2  at  the  Cambridge 
Congress. 

In  the  treatment  of  the  infinite  there  are  two  schools.  Georg  Cantor 
proved  that  the  continuum  is  not  denumerable;  J.  A.  Richard,  con- 
tending that  no  mathematical  entity  exists  that  is  not  definable  in  a 
finite  number  of  words,  argued  that  the  continuum  is  denumerable. 
H.  Poincare  claimed  that  this  contradiction  is  not  real,  since  J.  A. 
Richard  employs  a  non-predicative  definition.^  H.  Poincare,^  in 
discussing  the  logic  of  the  infinite,  states  that,  according  to  the  first 
school,  the  pragmatists,  the  infinite  flows  out  of  the  finite;  there  is  an 
infinite,  because  there  is  an  infinity  of  possible  finite  things.  Accord- 
ing to  the  second  school,  the  Cantorians,  the  infinite  precedes  the 
finite;  the  finite  is  obtained  by  cutting  off  a  small  piece  of  the  infinite. 

*  P.  E.  B.  Jourdain,  Contributions,  Chicago,  1915,  p.  206. 
"^  Bull.  Am.  Math.  Sac,  Vol.  17,  1911,  p.  193. 
^  Scientia,  Vol.  12,  1912,  pp.  i-n. 


ANALYSIS  403 

For  pragmatists  a  theorem  has  no  meaning  unless  it  can  be  verified; 
they  reject  indirect  proofs  of  existence;  hence  they  reply  to  E.  Zermelo 
who  proves  that  space  can  be  converted  into  a  well-ordered  aggregate 
(wohlgeordnete  Menge) :  Fine,  convert  it !  We  cannot  carry  out  this 
transformation  because  the  number  of  operators  is  infinite.  For 
Cantorians  mathematical  things  exist  independently  of  man  who  may 
think  about  them;  for  them  cardinal  number  is  no  mystery.  On  the 
other  hand,  pragmatists  are  not  sure  that  any  aggregate  has  a  cardinal 
number,  and  when  they  say  that  the  Machtigkeit  of  the  continuum 
is  not  that  of  the  whole  numbers,  they  mean  simply  that  it  is  im- 
]iossible  to  set  up  a  correspondence  between  these  two  aggregates, 
which  could  not  be  destroyed  by  the  creation  of  new  points  in  space. 
If  mathematicians  are  ordinarily  agreed  among  themselves,  it  is 
because  of  confirmations  which  pass  final  judgment.  In  the  logic  of 
infinity  there  are  no  confirmations. 

L.  E.  J.  Brouwcr  of  the  University  of  Amsterdam,  expressing  views 
of  G.  Mannoury,  said  in  1912  that  to  the  psychologist  belongs  the 
task  of  ex-plaining  "why  we  are  averse  to  the  so-called  contradictory 
systems  in  which  the  negative  as  well  as  the  positive  of  certain  propo- 
sitions are  valid,"  that  "the  intuitionist  recognizes  only  the  existence 
of  denumerable  sets"  and  "can  never  feel  assured  of  the  exactness  of 
mathematical  theory  by  such  guarantees  as  the  proof  of  its  being 
non-contradictory,  the  possibility  of  defining  its  concepts  by  a  finite 
number  of  words,  or  the  practical  certainty  that  it  will  never  lead  to 
misunderstanding  in  human  relations."  ^  A.  B.  Frizell  showed  in 
19 14  that  the  field  of  denumerably  infinite  processes  is  not  a  closed 
domain — a  concept  which  the  intuitionist  refuses  to  recognize,  but 
which  "need  not  disturb  an  intuitionist  who  cuts  loose  from  the  prin- 
cipium  contradictionis."  More  recent  tendencies  of  research  in  this 
field  are  described  by  E.  H.  Moore:  ^  "From  the  linear  continuum 
with  its  infinite  variety  of  functions  and  corresponding  singularities 
G.  Cantor  developed  his  theory  of  classes  of  points  {Punktmcngenlehre) 
with  the  notions:  limit-point,  derived  class,  closed  class,  perfect  class, 
etc.,  and  his  theory  of  classes  in  general  (allgemeine  Mengenlehre)  with 
the  notions:  cardinal  number,  ordinal  number,  order- type,  etc.  These 
theories  of  G.  Cantor  are  permeating  Modern  Mathematics.  Thus 
there  is  a  theory  of  functions  on  point-sets,  in  particular,  on  perfect 
])oint-sets,  and  on  more  general  order-t>pes,  while  the  arithmetic 
of  cardinal  numbers  and  the  algebra  and  function  theory  of  ordinal 
numbers  are  under  development. 

"Less  technical  generalizations  or  analogues  of  functions  of  the 
continuous  real  varial)le  occur  throughout  the  various  doctrines  and 
applications  of  analysis.  A  function  of  several  variables  is  a  function 
of  a  single  multipartite  varial)le;  a  distribution  of  potential  or  a  field 

1  L.  E.  J.  Brouwer  in  Bull.  A  m.  Math.  Soc,  Vol.  20,  1Q13,  pp.  84,  86. 

2  New  Haven  Colloquium,  igoO,  New  Haven,  19 10,  p.  2-4. 


404  A  HI  SI  OR  Y  OF  MATHEMATICS 

of  force  is  a  function  of  position  on  a  cuve  or  surface  or  region;  the 
value  of  the  definite  integral  of  the  Calculus  of  Variations  is  a  func- 
tion of  the  variable  function  entering  the  definite  integral;  a  curvilinear 
integral  is  a  function  of  the  path  of  integration;  a  functional  operation 
is  a  function  of  the  argument  function  or  functions;  etc.,  etc. 

"A  multipartite  variable  itself  is  a  function  of  the  variable  index 
of  the  part.  Thus  a  finite  sequence:  xi;  .  .  .;  x„,  of  real  numbers  is  a 
function  x  of  the  index  i,  viz.,  x{i)  =  X{  (i=i;  .  .  .;  n).  Similarly, 
an  infinite  sequence:  xi;  .  .  .;  x„;  .  .  .,  of  real  numbers  is  a  function  x 
of  the  index  n,  viz.,  x(n)=Xn  {n=i;  2;  .  .  .).  Accordingly,  «-fold 
algebra  and  the  theory  of  sequences  and  of  series  are  embraced  in 
the  theory  of  functions. 

"As  apart  from  the  determination  and  extension  of  notions  and 
theories  in  analogy  with  simpler  notions  and  theories,  there  is  the 
extension  by  direct  generalization.  The  Cantor  movement  is  in  this 
direction.  Finite  generalization,  from  the  case  w=i  to  the  case  w=^, 
occurs  throughout  Analysis,  as,  for  instance,  in  the  theory  of  func- 
tions of  several  independent  variables.  The  theory  of  functions  of  a 
denumerable  infinity  of  variables  is  another  step  in  this  direction.^ 
We  notice  a  more  general  theory  dating  from  the  year  1906.  Recog- 
nizing the  fundamental  role  played  by  the  notion  limit-element  (num- 
ber, point,  function,  curve,  etc.)  in  the  various  special  doctrines,  M. 
Frechet  has  given,  with  extensive  applications,  an  abstract  generaliza- 
tion of  a  considerable  part  of  Cantor's  theory  of  classes  of  points  and 
of  the  theory  of  continuous  functions  on  classes  of  points.  Frechet 
considers  a  general  class  P  of  elements  p  with  the  notion  limit  defined 
for  sequences  of  elements.  The  nature  of  the  elements  p  is  not  speci- 
fied; the  notion  limit  is  not  explicitly  defined;  it  is  postulated  as  de- 
fined subject  to  specified  conditions.  For  particular  applications 
explicit  definitions  satisfying  the  conditions  are  given.  .  .  .  The 
functions  considered  are  either  functions  /z  of  variables  p  of  specified 
character  or  functions  fj.  on  ranges  P  with  postulated  features:  e.  g. 
lifnit;  distance;  element  of  condensation;  connection,  of  specified  char- 
acter." E.  H.  Moore's  own  form  of  general  analysis  of  1906  considers 
functions  ix  of  a  general  variable  p  on  a  general  range  P,  where  this 
general  embraces  every  well-defined  particular  case  of  variable  and 
range. 

Early  in  the  development  of  the  theory  of  point  sets  it  was  pro- 
posed to  associate  with  them  numbers  that  are  analogous  to  those 
representing  lengths,  areas,  volumes.-  On  account  of  the  great  arbi- 
trariness of  this  procedure,  several  different  definitions  of  such  num- 
bers have  been  given.  The  earliest  were  given  in  1882  by  H.  Hankel 
and  A.  Harnack.  Another  definition  due  to  G.  Cantor  (1884)  was 
generalized  by  H.  Minkowski  in  1900.    More  precise  measures  were 

1  D.  Hilbert,  1906,  1909. 

^  Encyclopedis  des  sciences  niathettmliques,  Tome  II,  Vol.  i,  1912,  p.  150. 


ANALYSIS  405 

assigned  in  iSg^  by  C.  Jordan  in  his  Cours  d' analyse,  which,  lor  phine 
sets,  was  as  follows:  If  the  plane  is  divided  up  into  squares  whose 
sides  are  5  and  if  S  is  the  sum  of  the  squares  all  interior  points  of  which 
belong  to  the  set  F,  if,  moreover,  S+S'  is  the  sum  of  all  squares  which 
contain  points  of  P,  then,  as  5  a])proaches  zero,  5  and  S+S'  converge 
to  limits  /  and  A,  called  the  "interior"  and  "exterior"  areas.  If 
7=.l,  then  P  is  said  to  be  "measurable."  Examples  of  plane  curves 
whose  exterior  areas  were  not  zero  were  given  in  1903  by  W.  F.  Osgood 
and  H.  Lebesgue.  Another  definition,  given  by  E.  Borel,  w^as  gen- 
eralized by  H.  Lebesgue  so  as  to  present  fewer  inconveniences  than 
the  older  ones.  The  existence  of  non-measurable  sets,  according  to 
Lebesgue's  definition,  was  proved  by  G.  Vitali  and  Lebesgue  himself; 
the  proofs  assume  E.  Zermelo's  axiom. 

Instead  of  sets  of  points,  E.  Borel  in  1903  began  to  study  sets  of 
lines  or  planes;  G.  Ascoli  considered  sets  of  curves.  M.  Frechet  in 
1906  proposed  a  generalization  by  establishing  general  properties 
without  specifying  the  nature  of  the  elements,  and  was  led  to  the 
so-called  "calcul  fonctionnel,"  to  which  attention  has  been  called 
before. 

Functional  equations,  in  which  the  unknown  elements  are  one  or 
more  functions,  have  received  renewed  attention  in  recent  years.  In 
the  eighteenth  century  certain  tx-pes  of  them  were  treated  by  D'Alem- 
bert,  L.  Euler,  J.  Lagrange,  and  P.  S.  Laplace.^  Later  came  the 
"calculus  of  functions,"  studied  chiefly  by  C.  Babbage,  J.  F.  W. 
Herschel,  and  A.  De  Morgan,  which  was  a  theory  of  the  solution  of 
functional  equations  by  means  of  kno\Mi  functions  or  sjonbols.  A.  M. 
Legendre  and  A.  L.  Cauchy  studied  the  functional  equation /(.r-fy)  = 
f(x)+f(y);  which  recently  has  been  investigated  by  R.  Volpi,  G. 
Hamel,  and  R.  Schimmack.  Other  equations,  f{x+y)=f{x).f(y), 
f{xy)=f(x).f(y),  and  (p'y+x)+(p(y-x)  =  2(p(x).(p(y),  the  last  being 
D'Alembert's,  were  treated  by  A.  L.  Cauchy,  R.  Schimmack,  and 
J.  Andrade.  The  names  of  Ch.  Babbage,  N.  H.  Abel,  E.  Schroder,  J. 
Farkas,  P.  Appell,  and  E.  B.  Van  Vleck  are  associated  with  this  subject. 
Said  S.  Pincherele  of  Bologna  in  191 2:  "The  study  of  certain  classes 
of  functional  equations  has  given  rise  to  some  of  the  most  important 
chapters  of  analysis.  It  sufiQces  to  cite  the  theor>'  of  differential  or 
partial  differential  equations.  .  .  .  The  theory-  of  equations  of  finite 
differences,  simple  or  partial,  .  .  .  the  calculus  of  variations,  the 
theory  of  integral  equations  .  .  .  the  integrodifferential  equations 
recently  considered  by  V.  Volterra,  are  as  many  chapters  of  mathe- 
matics devoted  to  the  study  of  functional  equations." 

The  theory  of  point  sets  led  in  1902  to  a  generalization  of  G.  F.  B. 
Riemann's  definition  of  a  definite  integral  by  Henri  Lebesgue  of 
Paris.     E.  B.  Van  Vleck  describes  the  need  of  this  change;-  "This 

*  Encyclopedic  des  sciences  malhematiqiies ,  Tome  II,  Vol.  5,  1912,  p.  46. 
^Bull.  Am.  Math.  Soc,  Vol.  23, 19 16,  pp.  6,  7. 


4o6  A  HISTORY  OF  MATHEMATICS 

(Riemann's  integral)  admits  a  finite  number  of  discontinuities  but 
an  infinite  number  only  under  certain  narrow  restrictions.  A  totally 
discontinuous  function — for  example,  one  equal  to  zero  in  the  rational 
points  which  are  everyv\'here  dense  in  the  interval  of  integration, 
and  equal  to  i  in  the  rational  points  which  are  likewise  everywhere 
dense — is  not  integrable  a  la  Riemann.  The  restriction  became  a 
very  hampering  one  when  mathematicians  began  to  realize  that  the 
analytic  world  in  which  theorems  are  deducible  does  not  consist 
merely  of  highly  civilized  and  continuous  functions.  In  1902  Lebesgue 
with  great  penetration  framed  a  new  integral  which  is  identical  with 
the  integral  of  Riemann  when  the  latter  is  applicable  but  is  immeasur- 
ably more  comprehensive.  It  will,  for  instance,  include  the  totally 
discontinuous  function  above  mentioned.  This  new  integral  of 
Lebesgue  is  proving  itself  a  wonderful  tool.  I  might  compare  it  with 
a  modern  Krupp  gun,  so  easily  does  it  penetrate  barriers  which  before 
were  impregnable."  Instructive  is  also  the  description  of  this  move- 
ment, as  given  by  G.  A.  Bliss:  ^  "Volterra  has  pointed  out,  in  the 
introductory  chapter  of  his  Leqons  sur  les  Jonctions  des  lignes  (1913), 
the  rapid  development  which  is  taking  place  in  our  notions  of  infinite 
processes,  examples  of  which  are  the  definite  integral  limit,  the  solu- 
tion of  integral  equations,  and  the  transition  frcm  functions  of  a 
finite  number  of  variables  to  functions  of  lines.  In  the  field  of  in- 
tegration the  classical  integral  of  Riemann,  perfected  by  Darboux, 
was  such  a  convenient  and  perfect  instrument  that  it  impressed  itself 
for  a  long  time  upon  the  mathematical  public  as  being  something 
unique  and  final.  The  advent  of  the  integrals  of  T.  J.  Stieltjes  and  H. 
Lebesgue  has  shaken  the  complacency  of  mathematicians  in  this 
respect,  and,  with  the  theory  of  linear  integral  equations,  has  given 
the  signal  for  a  re-examination  and  extension  of  many  of  the  types  of 
processes  which  Volterra  calls  passing  from  the  finite  to  the  infinite. 
It  should  be  noted  that  the  Lebesgue  integral  is  only  one  of  the  evi- 
dences of  this  restlessness  in  the  particular  domain  of  the  integration 
theory.  Other  new  definitions  of  an  integral  have  been  devised  by 
Stieltjes,  W.  H.  Young,  J.  Pierpont,  E.  Hellinger,  J.  Radon,  M. 
Frechet,  E.  H.  Moore,  and  others.  The  definitions  of  Lebesgue, 
Young,  and  Pierpont,  and  those  of  Stieltjes  and  Hellinger,  form  two 
rather  well  defined  and  distinct  tj'pes,  while  that  of  Radon  is  a  gen- 
eralization of  the  integrals  of  both  Lebesgue  and  Stieltjes.  The 
efforts  of  Frechet  and  Moore  have  been  directed  toward  definitions 
valid  on  more  general  ranges  than  sets  of  points  of  a  line  or  higher 
spaces,  and  which  include  the  others  for  special  cases  of  these  ranges. 
Lebesgue  and  H.  Hahn,  with  the  help  of  somewhat  complicated 
transformations,  have  shown  that  the  integrals  of  Stieltjes  and  Hel- 

1  G.  A.  Bliss,  "Tnteerals  of  Lebesgue,"  Bull.  Am.  ilatJi.  Soc,  Vol.  24,  1917,  pp.  i- 
47.  See  also  T.  H.  Hildebiandt,  loc  ciL,  Vol.  24,  pp.  1 13-144,  who  gives  bibliogra- 
phy. 


ANALYSIS  407 

linger  are  expressible  as  Lebesgue  integrals.  .  .  .  Van  Vleck  has  .  .  . 
remarked  that  a  Lebesgue  integral  is  expressible  as  one  of  Stieltjes 
by  a  transformation  much  simpler  than  that  used  by  Lebesgue  fc)r 
the  opposite  puq:)ose,  and  the  Stieltjes  integral  so  obtained  is  readily 
expressible  in  terms  of  a  Riemann  integral.  .  .  .  Furthermore  the 
Stieltjes  integral  seems  distinctly  better  suited  than  that  of  Lebesgue 
to  certain  types  of  questions,  as  is  well  indicated  by  the  original 
'problem  of  moments'  of  Stieltjes,  or  by  a  generalization  of  it  which 
F.  Riesz  has  made.  .  .  .  The  conclusion  then  seems  to  be  that  one 
should  reserve  judgment,  for  the  present  at  least,  as  to  the  final  form 
or  forms  which  the  integration  theory  is  to  take." 

Mathematical  Logic 

Summarizing  the  history  of  mathematical  logic,  P.  E.  B.  Jourdain 
says:  ^  "In  somewhat  close  connection  with  the  work  of  Leibniz  .  .  . 
stands  the  work  of  Johann  Heinrich  Lambert,  who  sought — not  very 
successfully — to  develop  the  logic  of  relations.  Toward  the  middle 
of  the  nineteenth  century  George  Boole  independently  worked  out 
and  published  his  famous  calculus  of  logic.  .  .  .  Independently  of 
him  or  anybody  else,  Augustus  De  Morgan  began  to  work  out  logic 
as  a  calculus,  and  later  on,  taking  as  his  guide  the  maxim  that  logic 
should  not  consider  merely  certain  kinds  of  deduction  but  deduction 
quite  generally,  founded  all  the  essential  parts  of  the  logic  of  relations. 
William  Stanley  Jevons  criticised  and  popularized  Boole's  work;  and 
Charles  S.  Peirce  (1S39-1914),  Mrs.  Christine  Ladd-Franklin,  Richard 
Dedekind,  Ernst  Schroder  (1841-1902),  Hermann  G.  and  Robert 
Grassmann,  Hugh  MacCoU,  John  Venn,  and  many  others,  either 
developed  the  work  of  G.  Boole  and  A.  De  Morgan  or  built  up  systems 
of  calculative  logic  in  modes  which  were  largely  independent  of  the 
work  of  others.  But  it  was  in  the  work  of  Gottlob  Frege,  Guiseppe 
Peano,  Bertrand  Russell,  and  Alfred  North  Whitehead,  that  we  find 
a  closer  approach  to  the  lingua  character istica  dreamed  of  by  Leibniz." 
We  proceed  to  a  few  details, 

"Pure  mathematics,"  says  B.  Russell,^  "was  discovered  by  Boole 
in  a  work  which  he  called  The  Laws  of  Thought  (1854).  .  .  .  His 
work  was  concerned  with  formal  logic,  and  this  is  the  same  thing  as 
mathematics."  George  Boole  (18 15-1864)  became  in  1849  professor 
in  Queen's  College,  Cork,  Ireland.  He  was  a  native  of  Lincoln,  and 
a  self-educated  mathematician  of  great  power.  In  his  boyhood  he 
studied,  unaided,  the  classical  languages.^  While  teaching  school  he 
pursued  modem  languages  and  entered  upon  the  study  of  J.  Lagrange 

^  The  MonisI,  Vol.  26,  19 16,  p.  522. 

'^  Inlcrnalioiidl  Monthly,  1901,  p.  8v 

'See  A.  Macfarlane,  Ten  British  Mathematicians,  New  "\'()rk,  iqi6.  Boole's 
Laws  of  Thought  was  republished  in  19 17  by  tlie  Open  Court  Publ.  Co.,  under  the 
editorship  of  V.  E.  15.  Jourdain. 


4oS  A  HISTORY  OF  MATHEMATICS 

and  P.  S.  Laplace.  His  treatises  on  Differential  Equations  (1S59),  and 
Finite  Differences  (i860)  are  works  of  merit. 

A  point  of  view  different  from  that  of  G.  Boole  was  taken  by  Hugh 
MacColl  (183  7-1909)  who  was  led  to  liis  system  of  symbolic  logic  by 
researches  on  the  theory  of  probabiHty.  While  Boole  used  letters  to 
represent  the  times  during  which  certain  propositions  are  true,  Mac- 
Coll  employed  the  proposition  as  the  real  unit  in  symbolic  reasoning.^ 
When  the  variables  in  the  Boolean  algebra  are  interpreted  as  proposi- 
tions, C.  I.  Lewis  of  the  University  of  California  worked  out  a  matrix 
algebra  for  implications. 

WTien  the  investigation  of  the  principles  of  mathematics  became 
the  chief  task  of  logical  symbolism,  the  aspect  of  s\Tn.bolic  logic  as  a 
calculus  ceased  to  be  of  such  importance.  Friedrich  Ludwig  Gottlob 
Frege  (1S48-  )  of  the  University  of  Jena  entered  this  field.  Con- 
sidering the  foundations  of  arithmetic  he  inquired  how  far  one  could 
go  by  conclusions  which  rest  merely  on  the  laws  of  general  logic. 
Ordinary  language  was  found  to  be  unequal  to  the  accuracy  required. 
So  knowing  nothing  of  the  work  of  his  predecessors,  except  G.  W. 
Leibniz,  he  devised  a  symbolism  and  in  1879  published  his  Begriffs- 
schrift,  and  in  1S93  his  Grundgesetze  der  Arithmetik.  Says  P.  E.  B. 
Jourdain:  "Frege  criticised  the  notion  which  mathematicians  denote 
by  the  word  'aggregate'  (Menge),  and  particularly  the  views  of 
Dedekind  and  Schroder.  Neither  of  these  authors  distinguished  the 
subordination  of  a  concept  under  a  concept  from  the  falling  of  an 
object  under  a  concept;  a  distinction  upon  which  Peano  rightly  laid  so 
much  stress,  and  which  is,  indeed,  one  of  the  most  characteristic 
features  of  Peano's  system  of  ideography."  Ernst  Schroder  of  Karls- 
ruhe had  published  in  1S77  his  Algebra  der  Logik  and  John  Venn  his 
Symbolic  Logic  in  1S81. 

"Peano's  first  publication  on  mathematical  logic  followed  the 
lines  of  Schroder's  work  of  1S77  very  closely.  An  excellent  exposition 
in  Peano's  Calcolo  gco?netrico  secondo  VA  usdehmingslehre  di  H.  Grass- 
mann,  Turin,  1888,  of  the  geometrical  calculus  of  A.  F.  Mobius,  H.  G. 
Grassmann,  and  others  was  preceded  by  an  introduction  treating  of 
the  operations  of  deductive  logic,  which  are  very  analogous  to  those 
of  ordinary  algebra  and  of  the  geometrical  calculus.  The  signs  of 
logic  were  sometimes  used  in  the  later  parts  of  the  book,  though  this 
was  not  done  systematically,  as  it  was  in  many  of  Peano's  later  works" 
(Jourdain).  In  1891  appeared  under  G.  Peano's  editorship,  the  first 
volume  of  the  Rivista  di  Matematica  which  contains  articles  on  mathe- 
matical logic  and  its  applications,  but  this  kind  of  work  was  carried 
on  more  fully  in  the  Formiilaire  de  mathcmaiiques  of  which  the  first 
volume  was  published  in  1S95.  This  was  ]-)rojected  to  be  a  classified 
collection  of  mathematical  truths,  written  wholly  in  Peano's  symbols: 

1  For  details  see  Philip  E.  B.  Jourdain  in  Quarterly  Joiw.  of  Math.,  Vol.  43,  1912 
p.  2iy. 


A^AL^•S1S  409 

it  was  prepared  by  Pcano  and  his  collaborators,  C.  Eurali-Forti,  G. 
Viviania,  R.  Bettazzi,  F.  Giudice,  F.  Castellano,  and  G.  Fano.  "In 
the  later  editions  of  the  Formulaire,"  says  P.  E.  B.  Jourdain,  "Peano 
gave  up  all  attempts  to  wjrk  out  which  are  the  primitive  propositions 
of  logic;  and  the  logical  princi])les  or  theorems  which  are  used  in  the 
various  branches  of  mathematics  were  merely  collected  together  in 
as  small  a  space  as  possible.  In  the  last  edition  (v.,  1905),  logic  only 
occupied  16  pages,  while  mathematical  theories — a  fairly  complete 
collection— occui)ied  463  pages.  On  the  other  hand,  in  the  works  of 
Frege  and  B.  Russell  the  exact  enumeration  of  the  primitive  proposi- 
tions of  logic  was  always  one  of  the  most  important  problems."  In 
England  mathematical  logic  has  been  strongly  emphasized  by  Ber- 
trand  Russell  who  in  1903  published  his  Principles  of  Mathematics 
and  in  1910,  in  conjunction  with  A.  N.  Whitehead,  brought  out  the 
first  volume  of  the  Principia  malhematica,  a  remarkable  work.  Rus- 
sell and  Wliitehead  follow  in  the  main  Peano  in  matters  of  notation, 
Frege  in  matters  of  logical  analysis,  G.  Cantor  in  the  treatment  of 
arithmetic,  and  v.  Staudt,  M.  Pasch,  G.  Peano,  M.  Pieri,  and  O.  Veblen 
in  the  discussion  of  geometry.  By  their  theory  of  logical  i-ypes  they 
solve  the  paradoxes  of  C.  Burali-Forti,  B.  Russell,  J.  Konig,  J.  A. 
Richard,  and  others.  Certain  points  in  the  logic  of  relations  as  given 
in  the  Principia  mathematica  have  been  simplified  by  Norbert  Wiener 
in  1914  and  1915 

In  France  this  subject  has  been  cultivated  cliicfly  by  Louis  Couturat 
(1868-1914),  who,  at  the  time  of  his  death  in  an  automobile  accident 
in  Paris,  held  high  rank  in  the  philosophy  of  mathematics  and  of 
language.  He  wrote  La  logique  de  Leibniz  and  Lcs  principes  des  mathe- 
7natiques,  Paris,  1905.  Sets  of  postulates  for  the  Boolean  algebra  of 
logic  were  given  by  A.  N.  Whitehead,  which  were  simplified  in  1904 
by  E.  V.  Huntington  of  Harvard  and  B.  A.  Bernstein  of  California. 
Says  P.  E.  B.  Jourdain:  "Frege's  symboUsm,  though  far  better  for 
logical  analysis  than  Pet^i.o's,  for  instance,  is  far  inferior  to  Peano's — 
a  symbolism  in  which  the  merits  of  internationality  and  power  of 
expressing  mathv.matical  theorems  are  very  satisfactorily  attained— 
in  practical  convenience.  B.  Russell,  especially  in  the  later  works, 
used  the  ideas  of  Frege,  many  of  which  he  discovered  subsequently 
to,  but  independently  of  Frege,  and  modified  the  symbolism  of  Peano 
as  little  as  possible.  Still,  the  complications  thus  introduced  take 
away  that  simple  character  which  seems  necessary  to  a  calculus,  and 
which  Boole  and  others  reached  by  passing  over  certain  distinctions 
which  a  subtler  logic  has  shown  us  must  be  made."  ^ 

In  1886  A.  B.  Kempe  discussed  the  fundamental  concei)tions  both 
of  symbolic  logic  and  of  geometry.  Later  he  developed  this  subject 
further  in  the  study  of  the  relations  between  the  logical  theory  of 

'  Philij)  F,.  B.  Jourdain  in  Quart.  Jour,  of  y[alh.,  Vol.  41,  1910,  p.  331. 


4IO  A  HISTORY  OF  MATHEMATICS 

classes  and  the  geometrical  theory  of  points.  This  topic  received  a 
re-statement  and  an  extension  in  1905  from  the  pen  of  Josiah  Royce 
(1855-1916),  professor  of  philosophy  at  Harv^ard  University.  Royce 
contends  that  "the  entire  system  of  the  relationships  of  the  exact 
sciences  stands  in  a  much  closer  connection  with  the  simple  principles 
of  svmbolic  logic  than  has  thus  far  been  generally  recognized." 

There  exists  divergence  of  opinion  on  the  value  of  the  notation  of 
the  calculus  of  logic.  Says  A.  Voss:  '  "As  far  as  I  am  able  to  survey 
the  practical  results  of  mathematical  logic,  they  run  aground  at  every 
real  application,  on  account  of  the  extreme  complexity  of  its  formulas; 
by  a  comparatively  large  expenditure  of  effort  they  yield  almost 
trivial  results,  which,  however,  can  be  read  off  with  absolute  certainty. 
Only  in  the  discussion  of  purely  mathematical  questions,  i.  e.  relations 
between  numbers,  does  it,  in  Peano's  Formulaire  .  .  .,  prove  itself 
to  be  a  real  power,  probably  replaceable  by  no  other  mode  of  ex- 
pression.   By  some  even  this  is  called  into  question." 

Alessandro  Padoa  of  the  Royal  Technical  Institute  of  Genoa  said 
in  191 2:  -  "I  do  not  hope  to  suggest  to  you  the  sympathetic  and  touch- 
ing optimism  of  Leibniz,  who,  prophesying  the  triumphal  success  of 
these  researches,  affirmed:  'I  dare  say  that  this  is  the  last  effort  of 
the  human  mind,  and,  when  this  project  shall  have  been  carried  out, 
all  that  men  will  have  to  do  will  be  to  be  happy,  since  they  will  have 
an  instrument  that  will  serve  to  exalt  the  intellect  not  less  than  the 
telescope  serves  to  perfect  their  vision.'  Although  for  some  fifteen 
years  I  have  given  myself  up  to  these  studies,  I  have  not  a  hope  so 
hyperbolic;  but  I  delight  in  recalling  the  candor  of  this  master  who, 
absorbed  in  scientific  and  philosophic  investigations,  forgot  that  the 
majority  of  men  sought  and  continue  to  seek  happiness  in  the  feverish 
conquest  of  pleasure,  money,  and  honors.  Meanwhile  we  should 
avoid  an  excessive  scepticism,  because  always  and  everywhere,  there 
has  been  an  elite — to  day  less  restricted  than  in  the  past — which  was 
charmed  by,  and  delights  now  in,  all  that  raises  one  above  the  con- 
fused troubles  of  the  passions,  into  the  imperturbable  immensity  of 
knowledge,  whose  horizons  become  the  more  vast  as  the  wings  of 
thought  become  more  powerful  and  rapid." 

In  1914  an  international  congress  of  mathematical  philosophers  was 
held  in  Paris,  with  Emil  Boutroux  as  president.  Unfortunately  the 
great  war  nipped  this  promising  new  movement  in  the  bud.  Recent 
books  on  the  philosophy  of  mathematics  are  M.  Winter's  La  methode 
dans  la  philosophic  des  mathematics,  Paris,  191 1,  Leon  Brunschvicg's 
Les  ctapes  de  la  philosophic  mathematiquc,  P.iris,  191 2,  and  J.  B. 
Shaw's  Lectures  on  the  Philosophy  of  Mal.'irmct'cs,  C'licago  and  Lon- 
don, 191S. 

1  A.  Voss,  Ueber  das  Wesen  der  Alalhrmalik,  Leipzig  u.  Berlin,  2.  Aufl.,  1913, 
p.  28. 

2  Bull.  Am.  Math.  Soc,  Vol.  20,  1913,  p.  98. 


IHEORY  OF  FUNCTIONS  411 

Theory  of  Functions 

We  begin  our  sketch  of  the  vast  progress  in  the  theory  of  functions 
by  considering  investigations  which  center  about  the  special  class 
called  elliptic  functions.  These  were  richly  developed  by  N.  H.  Abei 
and  C.  G.  J.  Jacobi. 

Niels  Henrik  Abel  (1802-1829)  was  born  at  Findoe  in  Norway,  and 
was  prepared  for  the  university  at  tlie  cathedral  school  in  Christiania. 
He  exhibited  no  interest  in  mathematics  until  18 18,  when  B.  Holmboe 
became  lecturer  there,  and  aroused  Abel's  interest  by  assigning  original 
problems  to  the  class.  Like  C,  G.  J.  Jacobi  and  many  other  young 
men  who  became  eminent  mathematicians,  Abel  found  the  first  exer- 
cise of  his  talent  in  the  attempt  to  solve  by  algebra  the  general  equa- 
tion of  the  fifth  degree.  In  182 1  he  entered  the  University  in  Chris- 
tiania. The  works  of  L.  Euler,  J.  Lagrange,  and  A.  M.  Legendre 
were  closely  studied  by  him.  The  idea  of  the  inversion  of  elliptic 
functions  dates  back  to  this  time.  His  extraordinary  success  in 
mathematical  study  led  to  the  offer  of  a  stipend  by  the  government, 
that  he  might  continue  his  studies  in  Germany  and  France.  Leaving 
Norway  in  1825,  Abel  visited  the  astronomer,  H.  C.  Schumacher,  in 
Hamburg,  and  spent  six  months  in  Berlin,  where  he  became  intimate 
with  August  Leopold  Crelle  (1780-1855),  and  met  J.  Steiner.  En- 
couraged by  Abel  and  J.  Steiner,  Crelle  started  his  journal  in  1826. 
Abel  began  to  put  some  of  his  work  in  shape  for  print.  His  proof 
of  the  impossibility  of  solving  the  general  equation  of  the  fifth  degree 
by  radicals, — first  printed  in  1824  in  a  very  concise  form,  and  difficult 
of  apprehension, — was  elaborated  in  greater  detail,  and  published  in 
the  first  volume.  He  investigated  also  the  question,  what  equations 
are  solvable  by  algebra  and  deduced  important  general  theorems 
thereon.  These  results  were  published  after  his  death.  Meanwhile 
E.  Galois  traversed  this  field  anew.  Abel  first  used  the  expression, 
now  called  the  "Galois  resolvent";  Galois  himself  attributed  the  idea 
of  it  to  Abel.  Abel  showed  how  to  solve  the  class  of  equations,  now 
called  "Abelian."  Fie  entered  also  upon  the  subject  of  infinite  series 
(particularly  the  binomial  theorem,  of  which  he  gave  in  Crelle^s 
Journal  a  rigid  general  investigation),  the  study  of  functions,  and  of 
the  integral  calculus.  The  obscurities  everywhere  encountered  by  him 
owing  to  the  prevailing  loose  methods  of  analysis  he  endeavored  to 
clear  up.  For  a  short  time  he  left  Berlin  for  Freiberg,  where  he  had 
fewer  interruptions  to  work,  and  it  was  there  that  he  made  researches 
on  h>^erelliptic  and  Abelian  functions.  In  July,  1826,  Abel  left 
Germany  for  Paris  without  having  met  K.  F.  Gauss'  Abel  had  sent 
to  Gauss  his  proof  of  1824  of  the  impossibility  of  solving  equations  of 
the  fifth  degree,  to  which  Gauss  never  paid  any  attention.  This  slight, 
and  a  haughtiness  of  spirit  which  he  associated  with  Gauss,  prevented 
the  genie"  1  Abel  from  going  to  Gottingen.    A  similar  feeling  was  enter- 


412  A  HISTORY  OF  MATHEMATICS 

tained  by  him  later  against  A.  L.  Cauchy.  Abel  remained  ten  months 
in  Paris.  He  met  there  P.  G.  L.  Dirichlet,  A.  M.  Legendre,  A.  L. 
Cauchy,  and  others,  but  was  little  appreciated.  He  had  already  pub- 
lished several  important  memoirs  in  Crelle's  Journal,  but  by  the 
French  this  new  periodical  was  as  yet  hardly  known  to  exist,  and 
Abel  was  too  modest  to  speak  of  his  own  work.  Pecuniary  embarrass- 
ments induced  him  to  return  home  after  a  second  short  stay  in  Berlin. 
At  Christiania  he  for  some  time  gave  private  lessons,  and  served  as 
docent.  Crelle  secured  at  last  an  appointment  for  him  at  Berlin; 
but  the  news  of  it  did  not  reach  Norway  until  after  the  death  of 
Abel  at  Froland.^  Ch.  Hermite  is  said  to  have  remarked:  "Abel  a 
laisse  aux  mathematiciens  de  quoi  travailler  pendant  cent  cinquante 
ans." 

At  nearly  the  same  time  with  Abel,  C.  G.  J.  Jacobi  published  articles 
on  elliptic  functions,  A.  M.  Legendre's  favorite  subject,  so  long 
neglected,  was  at  last  to  be  enriched  by  some  extraordinary  discov- 
eries. The  advantage  to  be  derived  by  inverting  the  elliptic  integral 
of  the  first  kind  and  treating  it  as  a  function  of  its  amplitude  (now 
called  elliptic  function)  was  recognized  by  Abel,  and  a  few  months 
later  also  by  Jacobi.  A  second  fruitful  idea,  also  arrived  at  inde- 
pendently by  iDoth,  is  the  introduction  of  imaginaries  leading  to  the 
observation  that  the  new  functions  simulated  at  once  trigonometric 
and  exponential  functions.  For  it  was  shown  that  while  trigonometric 
functions  had  only  a  real  period,  and  exponential  only  an  imaginary, 
elliptic  functions  had  both  sorts  of  periods.  These  two  discoveries 
were  the  foundations  upon  which  Abel  and  Jacobi,  each  in  his  own 
way,  erected  beautiful  new  structures.  Abel  developed  the  curious 
expressions  representing  elliptic  functions  by  infinite  series  or  quo- 
tients of  infinite  products.  Great  as  were  the  achievements  of  Abel 
in  elliptic  functions,  they  were  eclipsed  by  his  researches  on  what  are 
now  called  Abelian  functions.  Abel's  theorem  on  these  functions  was 
given  by  him  in  several  forms,  the  most  general  of  these  being  that 
in  his  Memoire  siir  une  propriete  generate  d'une  classe  tres-etetidue  de 
Jonctions  transcendentes  (1826).  The  history  of  this  memoir  is  inter- 
esting. A  few  months  after  his  arrival  in  Paris,  Abel  submitted  it  to 
the  French  Academy.  A.  L.  Cauchy  and  A.  M.  Legendre  were  ap- 
pointed to  examine  it;  but  said  nothing  about  it  until  after  Abel's 
death.  In  a  brief  statement  of  the  discoveries  in  question,  published 
by  Abel  in  Crelle's  Journal,  1829,  reference  is  made  to  that  memoir. 
This  led  C.  G.  J.  Jacobi  to  inquire  of  Legendre  what  had  become  of  it. 
Legendre  says  that  the  manuscript  was  so  badly  written  as  to  be 
illegible,  and  that  Abel  was  asked  to  hand  in  a  better  copy,  which  he 

'  C.  A.  Bjerknes,  Niels-Henrik  Abel,  Tableau  de  sa  vie  et  de  son  action  scientifique, 
Paris,  1885.  See  also  Abel  {N.  H.)  Memorial  public  d  I' occasion  du  centenaire  de  sa 
naissance.  Kristiania  [1902];  also  N.  H.  Abel.  Sa  vie  et  son  Oettvre,  par  Ch.  Lucas  de 
Pesloiian,  Paris,  1906. 


THEORY  OF  FUNCTIONS  413 

neglected  to  do.  Others  have  attributed  this  failure  to  appreciate 
Abel's  paper  to  the  fact  that  the  French  academicians  were  then  in- 
terested chiefly  in  applied  mathematics — heat,  elasticity,  electricity, 
S.  D.  Poisson  having  in  a  report  on  C.  G.  J.  Jacobi's  Fiindamenta  nova 
recalled  the  reproach  made  by  J.  Fourier  to  Abel  and  Jacobi  of  not 
having  occupied  themselves  preferably  with  the  movement  of  heat, 
Jacobi  wrote  to  Legendre:  '"It  is  true  that  Monsieur  Fourier  held  the 
view  that  the  principal  aim  of  mathematics  was  public  utility,  and 
the  explanation  of  natural  phenomena;  but  a  philosopher  such  as  he 
should  have  known  that  the  unique  aim  of  science  is  the  honor  of  the 
human  spirit,  and  that  from  this  point  of  view  a  question  about 
numbers  is  as  important  as  a  question  about  the  system  of  the  world." 
In  1823  Abel  published  a  paper  ^  in  which  he  is  led,  by  a  mechanical 
question  including  as  a  special  case  the  problem  of  the  tautochrone, 
to  what  is  now  called  an  integral  equation,  on  the  solution  of  which 
the  solution  of  the  problem  depends.  His  problem  was,  to  determine 
the  curv^e  for  which  the  time  of  descent  is  a  given  function  of  the  ver- 
tical height.  In  \aew  of  the  recent  developments  in  integral  equa- 
tions, Abel's  problem  is  of  great  historical  interest.  Independently 
of  Abel,  researches  along  this  line  were  published  in  1832,  1837,  and 
1839  by  J.  Liouville,  who  in  1837  showed  that  a  particular  solution  of 
a  certain  differential  equation  can  be  obtained  by  the  aid  of  an  in- 
tegral equation  of  "  the  second  kind,"  somewhat  different  from  Abel's 
equation  "of  the  first  kind." 

Abel's  Manoire  of  1826  remained  in  A.  L.  Cauchy's  hands.  It  was 
not  pubHshed  until  1841.  By  a  singular  mishap,  the  manuscript  was 
lost  before  the  proof-sheets  were  read. 

In  its  form,  the  contents  of  the  memoir  belongs  to  the  integral 
calculus.  Abelian  integrals  depend  upon  an  irrational  function  y 
which  is  connected  with  .r  by  an  algebraic  equation  F(x,  y)=o.  Abel's 
theorem  asserts  that  a  sum  of  such  integrals  can  be  expressed  by  a 
definite  number  p  of  similar  integrals,  where  p  depends  merely  on  the 
properties  of  the  equation  F{x,  y)=o.  It  was  shown  later  that  p  is  the 
deficiency  of  the  curve  F(x,  y)=o.  The  addition  theorems  of  elliptic 
integrals  are  deducible  from_  Abel's  theorem.  The  hyperelliptic  in- 
tegrals introduced  by  Abel,  and  proved  by  him  to  possess  multiple 
periodicity,  are  special  cases  of  Abelian  integrals  whenever  p  =  ov  >3. 
The  reduction  of  Abelian  to  elliptic  integrals  has  been  studied  mainly  by 
C.  G.  J.  Jacobi,  Ch.  Hermite,  Leo  Konigsberger,  F.  Brioschi,  E.  Gour- 
sat,  E.  Picard,  and  O.  Bolza,  then  of  the  University  of  Chicago.  Abel's 
theorem  was  pronounced  by  Jacobi  the  greatest  discovery  of  our  cen- 
tury on  the  integral  calculus.  The  aged  Legendre,  who  greatly  ad- 
mired Abel's  genius,  called  it  "monumcntum  acre  pcrennius."  Some 
cases  of  Abel's  theorem  were  investigilcd  independently  by  William 
Henry  Fox  Talbot  (1S00-1877),  the  English  pioneer  of  photography, 
•  N  H.  Abel,  O^.uvrcs  completes,  18S1,  Vol.  i,  p.  11.    See  also  p.  97. 


414  A  HISTOR\  OF  MATHEMATICS 

who  showed  that  the  theorem  is  deducible  from  synunetric  functions 
of  the  roots  of  equations  and  partial  fractions.^ 

Two  editions  of  Abel's  works  have  been  published:  the  first  by 
Berndt  Michael  Holmboe  (i 795-1850)  of  Christiania  in  1839,  and  the 
second  by  L.  Sylow  and  S.  Lie  in  1881.  During  the  few  years  of  work 
allotted  to  the  young  Norwegian,  he  penetrated  new  fields  of  research. 
Abel's  published  papers  stimulated  researches  containing  certain 
results  previously  reached  by  Abel  himself  in  his  then  unpubhshed 
Parisian  memoir.  We  refer  to  papers  of  Christian  Jurgensen  (1805- 
1S61)  of  Copenhagen,  Ole  Jacob  Broch  (1818-1889)  of  Christiania, 
Ferdinand  Adolf  Minding  (1806-1S85)  of  Dorpat,  and  G.  Rosenhain. 

Some  of  the  discoveries  of  Abel  and  Jacobi  were  anticipated  by 
K.  F.  Gauss.  In  the  Disquisitiones  Arithmetics  he  observed  that 
the  principles  which  he  used  in  the  division  of  the  circle  were  ap- 
plicable to  many  other  functions,  besides  the  circular,  and  particularly 

C     dx 

to  the  transcendents  dependent  on  the  integral   I  —         .-.    From  this 

J  vi— .r 

Jacobi  -  concluded  that  Gauss  had  thirty  years  earlier  considered  the 
nature  and  properties  of  elliptic  functions  and  had  discovered  their 
double  periodicity.  The  papers  in  the  coUected  works  of  Gauss  con- 
firm this  conclusion. 

Carl  Gustav  Jacob  Jacobi  (1804-1S51)  was  born  of  Jewish  parents 
at  Potsdam.  Like  many  other  mathematicians  he  was  initiated  into 
mathematics  by  reading  L.  Euler.  At  the  University  of  Berhn,  where 
he  pursued  his  mathematical  studies  independently  of  the  lecture 
courses,  he  took  the  degree  of  Ph.D.  in  1825.  After  giving  lectures 
in  Berlin  for  two  years,  he  was  elected  extraordinary  professor  at 
Konigsberg,  and  two  years  later  to  the  ordinary  professorship  there. 
After  the  publication  of  his  Fundamenta  Nova  in  1829  he  spent  some 
time  in  travel,  meeting  Gauss  in  Gottingen,  and  A.  M.  Legendre, 
J.  Fourier,  S.  D.  Poisson,  in  Paris.  In  1842  he  and  his  colleague, 
F.  W.  Bessel,  attended  the  meetings  of  the  British  Association,  where 
they  made  the  acquaintance  of  English  mathematicians.  Jacobi 
was  a  great  teacher.  ''In  this  respect  he  was  the  very  opposite  of  his 
great  contemporary  Gauss,  who  disliked  to  teach,  and  who  was  any- ' 
thing  but  inspiring." 

Jacobi's  early  researches  were  on  Gauss'  approximation  to  the 
value  of  delinite  integrals,  partial  differential  equations,  Legendre's 
coefficients,  and  cubic  residues.  He  read  Legendre's  Exercises,  which 
give  an  account  of  elliptic  integrals.  When  he  returned  the  book  to 
the  library,  he  was  depressed  in  spirits  and  said  that  important  books 
generally  excited  in  him  new  ideas,  but  that  this  time  he  had  not 
been  led  to  a  single  original  thought.    Though  slow  at  first,  his  ideas 

'  G.  B.  Mathews  in  Nature,  Vol.  95,  19 15,  p.  21Q. 

-  R.  Tucker,  "Carl  Friedrich  Gauss,"  Nature,  April,  1877. 


THEORY  OF  FUNCTIONS 


41S 


flowed  all  the  richer  afterwards.  Many  of  his  discoveries  in  elliptic 
functions  were  made  independently  by  Abel.  Jacobi  communicated 
his  first  researches  to  Crelle's  Journal.  In  1829,  at  the  age  of  twenty- 
five,  he  published  his  Fundamenta  Nova  Theorice  Functionum  Ellfp- 
ticarum,  which  contains  in  condensed  form  the  main  results  in  elliptic 
functions.  This  work  at  once  secured  for  him  a  wide  reputation.  He 
then  made  a  closer  study  of  theta-functions  and  lectured  to  his  pupils 
on  a  new  theory  of  elliptic  functions  based  on  the  theta-functions.  He 
developed  a  theory  of  transformation  which  led  him  to  a  multitude 
of  formulre  containing  q,  a  transcendental  function  of  the  modulus, 
defined  by  the  equation  q  =  e~'^^  '*.  He  was  also  led  by  it  to  consider 
the  two  new  functions  H  and  G,  which  taken  each  separately  with 
two  ditTercnt  arguments  are  the  four  (single)  theta-functions  desig- 
nated by  the  9i,  02,  Qs,  04.^  In  a  short  but  very  important  memoir 
of  1833,  he  shows  that  for  the  h}q)erelliptic  integral  of  any  class  the 
direct  functions  to  which  Abel's  theorem  has  reference  are  not  func- 
tions of  a  single  variable,  such  as  the  elliptic  sn,  en,  dn,  but  functions 
of  p  v?nables.^  Thus  in  the  case  p=2,  which  Jacobi  especially  con- 
siders, it  is  shown  that  Abel's  theorem  has  reference  to  two  functions 
\{u,  v),  \\{h,  v),  each  of  two  variables,  and  gives  in  effect  an  addition- 
theorem  for  the  expression  of  the  functions  \{u-\-u' ,  v-\-v'),  \](u+u', 
v+v')  algebraically  in  terms  of  the  functions  \(tc,v),  Xi(u,v),  X{u',v'), 
X](u',  v')-  By  the  memoirs  of  N.  H.  Abd  and  Jacobi  it  may  be  con- 
sidered that- the  notion  of  the  Abilian  function  of  p  variables  was 
established  and  the  addition-theorem  for  these  functions  given.  Re- 
cent studies  touching  Abelian  functions  have  been  made  by  K.  Weier- 
strass,  E.  Picard,  Madame  Kovalevski,  and  H.  Poincare.  Jacobi's 
work  on  differential  equations,  determinants,  dynamics,  and  the 
theory  of  numbers  is  mentioned  elsewhere. 

In  1842  C.  G.  J.  Jacobi  visited  Italy  for  a  few  months  to  recuperate 
his  health.  At  this  time  the  Prussian  government  gave  him  a  pension, 
and  he  moved  to  Berlin,  where  the  last  years  of  his  life  were  spent. 

Among  those  who  greatly  extended  the  researches  on  functions 
mentioned  thus  far  was  Charles  Hermite  (1822-1901),  who  was  born 
at  Dieuze  in  Lorraine.-  He  early  manifested  extraordinary  talent  for 
mathematics.  Neglecting  the  regular  courses  of  study,  he  read  in 
Paris  with  greatest  ardor  the  masterpieces  of  L.  Euler,  J.^Lagrange, 
K.  F.  Gauss,  and  C.  G.  J.  Jacobi.  In  1842  he  entered  the  Ecole  Poly- 
technique.  From  birth  he  had  suffered  from  an  infirmity  of  the  right 
leg  and  had  to  use  a  cane.  On  this  account  he  was  declared  ineligible 
to  any  government  position  given  to  graduates  of  the  Ecole.  Hermite, 
therefore,  left  at  the  end  of  the  first  year.  A  letter  to  Jacobi  displayed 
his  mathematical  genius,  but  the  necessity  of  taking  examinations 
which  he  held  en  horreiir  compelled  him  to  descend  from  his  lofty 

'  \rthur  Cayley,  Inaugural  Address  before  the  British  Association,  1883. 

2  Bull.  Am.  Math.  Soc,  Vol.  13,  1907,  p.  182. 


416  A  HISTORY  OF  MATHEMATICS 

mathematical  speculations  and  take  up  the  irksome  details  prepara- 
tory to  examinations.  In  1848^ he  became  examinateur  d'admission 
and  repetiteur  d'analyse  at  the  Ecole  Polytechnique.  In  that  position 
he  succeeded  P.  L.  Wantzel.  That  year  he  married  a  sister  of  his 
friend,  Joseph  Bertrand.  In  1S69,  at  the  age  of  forty-seven,  he  became 
professor  and  at  length  reached  a  position  befitting  his  talents.  At 
the  Sorbonne  he  succeeded  J.  M.  C.  Duhamel  as  professor  of  higher 
algebra.  He  occupied  the  chair  at  the  Ecole  Polytechnique  until 
1S76,  at  the  Sorbonne  until  1897.  For  many  years  he  had  been  re- 
garded as  the  venerated  chief  among  French  mathematicians.  Hermite 
had  no  fondness  for  geometry.  His  researches  are  confined  to  algebra 
and  analysis.  He  wrote  on  the  theory  of  numbers,  invariants  and 
covariants,  definite  integrals,  theory  of  equations,  elliptic  functions 
and  the  theory  of  functions.  Of  his  collected  works,  or  Oeuvres, 
Vol.  Ill  appeared  in  191 2,  edited  by  E.  Picard.  In  the  theory  of 
functions  he  was  the  foremost  French  writer  of  his  day,  since  A.  L. 
Cauchy.  He  has  given  an  entirely  new  significance  to  the  use  of 
definite  integrals  in  the  theory  of  functions:  we  name  the  develop- 
ments of  the  properties  of  the  gamma-function  which  have  been  thus 
initiated. 

Elliptic  functions,  considered  on  the  Jacobian  rather  than  on  the 
Weierstrassian  basis,  was  a  favorite  study  of  Hermite.  "To  him  is 
due  the  reduction  of  an  elliptic  integral  to  its  canonical  form  by  means 
of  the  syzygy  among  the  concomitants  of  a  binary  quartic.  His  in- 
vestigations on  modular  functions  and  modular  equations  are  of  the 
highest  importance.  It  was  Hermite  who  discovered  pseudo-periodic 
functions  of  the  second  kind,  and  developed  their  properties.  In  a 
memoir  that  may  be  fairly  described  as  classical,  '  Sur  quelques  appli- 
cations des  fonctions  elliptiques'  in  the  Comptes  Rendus,  1877-1882, 
he  applied  these  functions  to  the  integration  of  the  unspecialized  form 
of  Lame's  differential  equation;  and  elliptic  functions  generally  were 
applied  in  that  memoir  to  obtain  the  solution  of  a  number  of  physical 
problems"  (A.  R.  Forsyth). 

In  1858  Hermite  introduced  in  place  of  the  variable  q  of  Jacobi  a 
new  variable  w  connected  with  it  by  the  equation  q=e^'^^,  so  that  w= 
ik'lk,  and  was  led  to  consider  the  functions  ^(w),  '/'(w),  x(^)-^ 
Henry  Smith  regarded  a  theta-f unction  with  the  argument  equal  to 
zero,  as  a  function  of  w.  This  he  called  an  omega-function,  while 
the  three  functions  <^(w),  </'(<o),  x('")>  ^-^e  his  modular  functions. 
Researches  on  theta-functions  with  respect  to  real  and  imaginary 
arguments  have  been  made  by  Ernst  Meissel  (1826-1895)  of  Kiel, 
J.  Thomae  of  Jena,  Alfred  Enneper  (1830-1885)  of  Gottingen.  A 
general  formula  for  the  product;  of  two  theta-functions  was  given  in 
1854  by  H.  Schroter  (1829-1892)  of  Breslau.    These  functions  have 

1  Arthur  Cayley,  Inaugural  Address,  1883. 


rilEOkV  Ol'    Fl^NC^TIONS  417 

been  studicfl  also  by  Cauchy,  Kiinip^sbergcr  of  Heidelberg  (born  1837), 
Friedrich  Julius  Richelot  (t8o8-iS75)  of  Konigsberg,  Johann  Georg 
Rosenhain  (1816-1887)  of  Konigsberg,  Ludwig  Schlafli  (1814-1895) 
of  Bern.^ 

A.  M.  Legendre's  method  of  reducing  an  elliptic  differential  to  its 
normal  form  has  called  forth  many  investigations,  most  important 
of  which  are  those  of  F.  J.  Richelot  and  of  K.  Weierstrass  of  BerUn. 

The  algebraic  transformations  of  elliptic  functions  involve  a  relation 
between  the  old  modulus  and  the  new  one  which  C.  G.  J.  Jacobi  ex- 
pressed by  a  differential  equation  of  the  third  order,  and  also  by  an 
algebraic  equation,  called  by  him  "modular  equation."  The  notion 
of  modular  equations  was  familiar  to  Abel,  but  the  development  of 
this  subject  devolved  upon  later  investigators.  These  equations 
have  become  of  importance  in  the  theory  of  algebraic  equations,  and 
have  been  studied  by  Ludwig  Adolph  Sohncke  (1807-1853)  of  Halle, 
E.  Mathieu,  L.  Konigsberger,  E.  Betti  of  Pisa,  Ch.  Hermite  of  Paris, 
P.  Joubert  of  Angers,  Francesco  Brioschi  of  Milan,  L.  Schlafli,  H. 
Schroter,  C.  Gudermann  of  Cleve,  Carl  Eduard  Giitzlaff  (1805-?)  of 
Marienwerder  in  Prussia. 

Felix  Klein  of  Gottingen  has  made  an  extensive  study  of  modular 
functions,  dealing  with  a  type  of  operations  lying  between  the  two 
extreme  ty]jes,  known  as  the  theory  of  substitutions  and  the  theory 
of  invariants  and  covariants.  Klein's  theory  has  been  presented  in 
book-form  by  his  pupil,  Robert  Fricke.  The  bolder  features  of  it 
wore  first  published  in  his  Ikosaeder,  1884.  His  researches  embrace 
the  theory  of  modular  functions  as  a  specific  class  of  elliptic  fmictions, 
the  statement  of  a  more  general  problem  as  based  on  the  doctrine 
of  groups  of  operations,  and  the  further  development  of  the  subject 
in  connection  with  a  class  of  Riemann's  surfaces. 

The  elliptic  functions  were  expressed  by  N.  H.  Abel  as  quotients 
of  doubly  infinite  products.  He  did  not,  however,  inquire  rigorously 
into  the  convergency  of  the  products.  In  1S45  A.  Cayley  studied 
these  products,  and  found  for  them  a  complete  theory,  based  in  part 
upon  geometrical  interpretation,  which  he  made  the  basis  of  the  whole 
theory  of  elliptic  functions.  F.  G.  Eisenstein  discussed  by  purely 
analytical  methods  the  general  doubly  infinite  product,  and  arrived 
at  results  which  have  been  greatly  simplified  in  form  by  the  theor}.'  of 
primary  factors,  due  to  K.  Weierstrass.  A  certain  function  involving 
a  doubly  infinite  ])roduct  has  been  called  by  Weierstrass  the  sigma- 
function,  and  is  the  basis  of  his  beautiful  theory  of  elliptic  functions. 
The  first  systematic  presentation  of  Weierstrass'  theory  of  elliptic 
functions  was  published  in  1886  by  G.  H.  Halphen  in  his  Theorie  des 
fondions  dliptiques  et  des  leurs  applications.  Applications  of  these 
functions  have  been  given  also  by  A.  G.  Greenhill  of  London.    Gener- 

1  .\lfred  Enncp(.T,  F.l  iptis  lie  Fuiiklioncn.  Thcoric  unci  Gcscliichlc.  Ilallc  a 'S,  1876. 


4i8  A  HISTORY  OF  MATHEMATICS 

alizations  analogous  to  those  of  Weierstrass  on  elliptic  functions  have 
been  made  by  Felix  Klein  on  hy|Derelliptic  functions. 

Standard  works  on  elliptic  functions  have  been  pubHshed  by  C.  A. 
A.  Briot  and  /.  C.  Bouquet  (1859),  by  L.  Konigsberger,  A.  Cayley, 
Heinrich  Durege  (1821-1893)  of  Prague,  and  others. 

Jacobi's  work  on  Abelian  and  theta-functions  was  greatly  extended 
by  Adolph  Gopel  (i8i2)-i847),  professor  in  a  g^'^mnasium  near  Pots- 
dam, and  Johaiin  Georg  Rosenhain  (1816-18S7)  of  Konigsberg. 
Gopel  in  his  Thcor'uv  transcendentiiim  primi  ordinis  adunibratio  Icvis 
{Crellc,  35,  1847)  3.nd  Rosenhain  in  several  memoirs  established  each 
independently,  on  the  analogy  of  the  single  theta-functions,  the  func- 
tions of  two  variables,  called  double  theta-functions,  and  worked  out 
in  connection  with  them  the  theory  of  the  Abelian  functions  of  two 
variables.  The  theta-relations  established  by  Gopel  and  Rosenhain 
received  for  thirty  years  no  further  development,  notwithstanding 
the  fact  that  the  double  theta  series  came  to  be  of  increasing  impor- 
tance in  analytical,  geometrical,  and  mechanical  problems,  and  that 
Ch.  Hermite  and  L.  Konigsberger  had  considered  the  subject  of  trans- 
formation. Finally,  the  investigations  of  C.  W.  Borchardt,  treating 
of  the  representation  of  Rummer's  surface  by  Gopel's  biquadratic 
relation  between  four  theta-functions  of  two  variables,  and  researches 
of  H.  H.  Weber,  F.  Prym,  Adolf  Krazer,  and  Martin  Krause  of  Dres- 
den led  to  broader  views.  Carl  Wilhelm  Borchardt  (18 17-1880)  was 
born  in  Berlin,  studied  under  P.  G.  L.  Dirichlet  and  C.  G.  J.  Jacobi 
in  Germany,  and  under  Ch.  Hermite,  M.  Chasles,  and  J.  Liouville 
in  France.  He  became  professor  in  Berlin  and  succeeded  A.  L.  Crelle 
as  editor  of  the  Journal  fur  Mathematik.  Much  of  his  time  was  given 
to  the  applications  of  determinants  in  mathematical  research. 

Friedrich  Prym  (1841-1915)  studied  at  Berlin,  Gdttingen,  and  Hei- 
delberg. He  became  professor  at  the  Polytechnicum  in  Zurich,  then  at 
Wlirzburg.  His  interest  lay  in  the  theory  of  functions.  Researches 
on  double  theta-functions,  made  by  A.  Cayley,  were  extended  to 
quadruple  theta-functions  by  Thomas  Craig  (1855-1900),  professor 
at  the  Johns  Hopkins  University.  He  was  a  pupil  of  J.  J.  Sylvester. 
While  lecturing  at  the  University  he  was  during  1879-1881  connected 
with  the  United  States  Coast  and  Geodetic  Survey.  For  many  years 
he  was  an  editor  of  the  American  Journal  of  Mathematics. 

Starting  with  the  integrals  of  the  most  general  form  and  considering 
the  inverse  functions  corresponding  to  these  integrals  (the  Abelian 
functions  of  p  variables),  G.  F.  B.  Riemann  defined  the  theta-functions 
of  p  variables  as  the  sum  of  a  />-tuply  infinite  series  of  exponentials, 
the  general  term  depending  on  p  variables.  Riemann  shows  that  the 
Abelian  functions  are  algebraically  connected  with  theta-functions  of 
the  proper  arguments,  and  presents  the  theory  in  the  broadest  form/ 

*  Arthur  Cayley,  Inaugural  Address,  1883. 


THEORY  OF  FUNCTIONS  419 

He  rests  the  theory  of  the  multiple  theta-f unctions  upon  the  general 
principles  of  the  theory  of  functions  of  a  complex  variable. 

Through  the  researches  of  A.  Brill  of  Tubingen,  M.  Nother  of 
Erlangen,  and  Ferdinand  Lindemann  of  Munich,  made  in  connection 
with  Riemann-Roch's  theorem  and  the  theory  of  residua tion,  there 
has  grown  out  of  the  theory  of  Abelian  functions  a  theory  of  algebraic 
functions  and  point-groups  on  algebraic  curves. 

General  Theory  of  Functions 

The  histor>'  of  the  general  theory  of  functions  begins  with  the 
adoption  of  new  definitions  of  a  function.  As  an  inheritance  from  the 
eighteenth  century,  y  was  called  a  function  of  x,  if  there  existed  an 
equation  between  these  variables  which  made  it  possible  to  calculate 
y  for  any  given  value  of  x  lying  anywhere  between  —  co  and  +  00 . 
We  have  seen  that  L.  Euler  sometimes  used  a  second,  more  general, 
definition,  which  was  adopted  by  J.  Fourier  and  which  was  translated 
by  P.  G.  L.  Dirichlet  into  the  language  of  analysis  thus:  y  is  called  a 
function  of  x,  if  y  possess  one  or  more  definite  values  for  each  of  cer- 
tain values  that  x  is  assumed  to  take  in  an  interval  xo  to  xi.  In  func- 
tions thus  defined,  there  need  be  no  analytical  connection  between 
y  and  x,  and  it  becomes  necessary  to  look  for  possible  discontinuities. 
This  definition  was  still  further  emphasized  and  generalized  later, 
after  the  introduction  of  the  theory  of  aggregates.  There  a  function 
need  not  be  defined  'for  each  point  in  the  continuum  embracing  all 
real  and  complex  numbers,  nor  for  each  point  in  an  interval,  but  only 
for  the  points  x  in  some  particular  set  of  points.  Thus,  y  is  a  function 
of  .V,  if  for  each  point  or  number  in  any  set  of  points  or  numbers  x, 
there  corresponds  a  point  or  number  in  a  set  y. 

P.  G.  L.  Dirichlet  lectured  on  the  theory  of  the  potential  and  thereby 
made  this  theory  more  generally  known  in  Germany.  In  1839  K.  F. 
Gauss  had  made  researches  on  the  potential;  in  England  George 
Green  had  issued  his  fundamental  memoir  as  early  as  1828.  Dir- 
ichlet's  lectures  on  the  potential  became  knowm  to  G.  F.  B.  Riemann 
who  made  it  of  fundamental  importance  for  the  whole  of  mathe- 
matics. Before  considering  Riemann  we  must  take  up  A.  L. 
Cauchy. 

J.  Fourier's  declaration  that  any  given  arbitrary  function  can  be 
represented  by  a  trigonometric  series  led  Cauchy  to  a  new  formulation 
of  the  concepts  " continuous,"  "limiting  value "  and  "function."  In  his 
Cours  (TAnalyse,  1821,  he  says:  "The  function  /(x)  is  continuous 
between  two  given  limits,  if  for  each  value  of  x  that  lies  between 
these  limits,  the  numerical  value  of  the  difference  /(x-|-  a)  -fix)  di- 
minishes with  a  in  such  a  way  as  to  become  less  than  every  finite 
number  "  (Chap.  II,  §  2).  With  S.  F.  Lacroix  and  A.  L.  Cauchy  there 
are  indications  of  a  tendency  to  free  the  functional  concept  from  an  ac- 


420  A  HISTORY  OF  MATHEMATICS 

tual  representation.^  Although  in  his  earlier  UTitings  slow  to  recognize 
the  importance  of  imaginary  variables,  Cauchy  later  entered  deeply 
into  the  treatment  of  functions  of  complex  variables,  not  in  a  geometri- 
cal form  as  found  in  C.  Wessel,  J.  R.  Argand,  and  K.  F.  Gauss,  but 
rather  in  analytical  form.  He  carried  on  integrations  through  imag- 
inary fields.  While  L.  Euler  and  P.  S.  Laplace  had  declared  the  order 
of  integration  in  double  integrals  to  be  immaterial,  A.  L.  Cauchy 
showed  that  this  was  true  only  when  the  expression  to  be  integrated 
does  not  become  indeterminate  in  the  interval  {Memoire  sur  la  theorie 
des  integrates  dcfiiiies,  read  1814,  printed  1825). 

If  between  two  paths  of  integration,  in  the  complex  plane,  there 
lies  a  pole,  then  the  diilerence  between  the  respective  integrals  can  be 
represented  by  means  of  a  "residu  de  la  fonction"  (1826),  a  concept  of 
undoubted  importance  known  as  the  calculus  of  residues.  In  1846 
he  showed  that  if  X  and  Y  are  continuous  functions  of  x  and  y  within 

a   closed   area,   then    I  (A'<f.v+Frfy)  =  ±   I    /( jdxdy,  where 

the  left  integral  extends  over  the  boundary  and  the  right  integral 
over  the  inner  area  of  the  complex  plane;  he  considers  integration 
along  a  closed  path  surrounding  a  "pole,"  and  later  along  a  closed 
path  surrounding  a  line  on  which  the  function  is  discontinuous,  as 
for  instance  log  x  for  x<o  when  the  function  changes  by  2Tri  in  crossing 
the  X-axis.  The  fundamental  theorem  of  Cauqhy's  theory  of  series 
was  given  in  1837 :  "A  function  can  be  expanded  in  an  ascending  power 
series  in  x,  as  long  as  the  modulus  of  x  is  less  than  that  for  which  the 
function  ceases  to  be  finite  and  continuous."  In  1840  the  proof  of  this 
theorem  is  made  to  rest  on  the  theorem  of  mean  value.  Cauchy, 
J.  C.  F.  Sturm,  and  J.  Liouville  had  carried  on  discussions  as  to 
w^hether  the  continuity  of  a  function  was  sufficient  to  insure  its  ex- 
pandibility  or  whether  that  of  its  derivative  must  be  demanded  as  well. 
In  1851  Cauchy  concluded  that  the  continuity  of  the  derivative  must 
be  demanded.  A  function  f(z),  which  is  single-valued  for  z=x+iy 
was  called  by  Cauchy  "monotyi^ique,"  later  "monodrome,"  by  Briot 
and  Bouquet  "monotrope,"  by  Hermite  "uniforme,"  by  the  Germans 
"eindeutig."  Cauchy  called  a  function  "monogen"  when  for  every 
s  in  a  region  it  had  only  one  derivative  value,  "synectique"  if  it  is 
monodromic  and  monogenic  and  does  not  become  infinite.  Instead 
of  "synectique,"  C.  A.  A.  Briot  and  J.  C.  Bouquet,  and  later  French 
writers  say  "holomorph,"  also  "meromorph"  when  the  function  has 
"poles"  in  the  region. 

Some  parts  of  Cauchy's  theory  of  functions  were  elaborated  by 
P.  M.  H.  Laurent  and  Victor  Alexandre  Puiseux  (1820-1883),  both 

1  A.  Brill  und  M.  Noether,  "Entwicklung  der  Theorie  der  algebraischen  Func- 
tionen  in  alterer  und  neucrer  Zeit,"  Jahresb.  d.  d.  Math.  Vereinig.,  Vol.  3,  1892-1893 
p.  162.    We  are  making  extensive  use  of  this  historical  monograph. 


THEORY  OF  FUXCTIONS  421 

of  Paris.  Laurent  pointed  out  the  advantage  resulting  in  certain 
cases  from  a  mixed  expansion  in  ascending  and  descending  powers  of  a 
variable,  while  Puiseux  demonstrated  the  advantage  that  may  be 
gained  by  the  use  of  series  involving  fractional  powers  of  the  variable. 
Puiseux  examined  many-valued  algebraic  functions  of  a  complex  va- 
riable, their  branch-points  and  moduli  of  periodicity. 

We  proceed  to  investigations  made  in  Germany  by  G.  F.  B. 
Riemann. 

Georg  Friedrich  Bernhard  Riemann  (1826-1866)  was  born  at 
Breselcnz  in  Hanover.  His  father  wished  him  to  study  theology,  and 
he  accordingly  entered  upon  philological  and  theological  studies  at 
Gottingen.  He  attended  also  some  lectures  on  mathematics.  Such 
was  his  predilection  for  this  science  that  he  abandoned  theology. 
After  studying  for  a  time  under  K.  F.  Gauss  and  M.  A.  Stern,  he 
was  dri-wn,  in  1847,  to  Berlin  by  a  galaxy  of  mathematicians,  in 
which  shcne  P.  G.  L.  Dirichlet,  C.  G.  J.  Jacobi,  J.  Steiner,  and  F.  G. 
Eisenstein.  Returning  to  Gottingen  in  1850,  he  studied  physics  under 
W.  Weber,  and  obtained  the  doctorate  the  following  year.  The  thesis 
presented  on  that  occasion,  Griindlagen  jur  eine  allgemcine  Theorie  der 
Funktionen  einer  ver'dnderlichen  complexen  Grosse,  excited  the  admira- 
tion of  K.  F.  Gauss  to  a  very  unusual  degree,  as  did  also  Riemann's 
trial  lecture,  Ucber  die  Ilypothesen  welche  dcr  Geornetrie  zu  Grunde 
liegen.  Influenced  by  Gauss  and  W.  Weber,  physical  views  were  the 
mainspring  of  his  purely  mathematical  investigations.  Riemann's 
Habilitationsschrift  (1854,  published  1867)  was  on  the  Representa- 
tion of  a  Function  by  means  of  a  Trigonometric  Series,  in  which  he 
advanced  materially  beyond  the  position  of  Dirichlet.  A.  L.  Cauchy 
had  set  up  criteria  for  the  existence  of  a  definite  integral  defined  as 
the  limit  of  a  sum,  and  had  stated  that  such  a  limit  always  exists 
when  the  function  is  continuous.  Riemann  made  a  starthng  extension 
by  pointing  out  that  the  existence  of  such  a  limit  is  not  confined  to 
cases  of  continuity.  Riemann's  new  criterion  placed  the  definite 
integral  upon  a  foundation  wholly  independent  of  the  differential 
calculus  and  the  existence  of  a  derivative.  It  led  to  the  consideration 
of  areas  and  lengths  of  arcs  which  may  transcend  all  geometric  figures 
within  the  reach  of  our  intuitions.  Half  a  century  later  the  concept 
of  a  definite  integral  was  still  further  extended  by  H.  Lebesgue  of 
Paris  and  others.  Our  hearts  are  drawn  to  Riemann,  an  extraordina- 
rily gifted  but  shy  genius,  when  we  read  of  the  timidity  and  nerv'ousness 
displayed  when  he  began  to  lecture  at  Gottingen,  and  of  his  jubilation 
over  the  unexpectedly  large  audience  of  eight  students  at  his  first 
lecture  on  differential  equations. 

Later  he  lectured  on  Abelian  functions  to  a  class  of  three  only, 
E.  C.  J.  Schering,  Bjerknes,  and  Uedekind.  K.  F.  Gauss  died  in  1855, 
and  was  succeeded  by  P.  G.  L.  Dirichlet.  On  the  death  of  the  latter, 
in  1859,  Riemann  was  mad..'  ordinary  professor.     In  i860  he  visited 


422  A  HISTORY  OF  MATHEMATICr, 

Paris,  where  he  made  the  acquaintance  of  French  mathematicians. 
The  delicate  state  of  his  health  induced  him  to  go  to  Italy  three  times. 
He  died  on  his  last  trip  at  Selasca,  and  was  buried  at  Biganzolo. 

Like  all  of  Riemann's  researches,  those  on  functions  were  profound 
and  far-reaching.  A  decidedly  modern  tendency  was  his  mode  of  in- 
vestigating functions.  In  the  words  of  E.  B.  Van  Vleck:  ^  "He  [Rie- 
mann]  presents  a  strange  antithesis  to  his  contemporary  countryman, 
Weierstrass.  Riemann  bases  the  function  theory  upon  a  property 
rather  than  upon  an  algorism — to  wit,  the  possession  of  a  differential 
coefficient  by  the  function  in  the  complex  plane.  Thus  at  a  stroke 
it  is  freed  from  dependence  upon  a  particular  process  like  the  power 
series  of  Taylor.  His  celebrated  memoir  upon  the  P-function  is  a 
characteristic  development  of  a  whole  Schar  (family)  of  functions 
from  their  mutual  relations." 

G.  F.  B.  Riemann  laid  the  foundation  for  a  general  theory  of  func- 
tions of  a  complex  variable.  The  theory  of  potential,  which  up  to 
that  time  had  been  used  only  in  mathematical  physics,  was  applied 
by  him   in   pure   mathematics.     He   accordingly  based  his  theory 

of  functions  on    the   partial  differential  equation,   — ^H 7.=^u  =  o, 

which  must  hold  for  the  analytical  function  w=u+h  of  z=x+iy. 
It  had  been  proved  by  P.  G.  L.  Dirichlet  that  (for  a  plane)  there  is 
always  one,  and  only  one,  function  of  x  and  y,  which  satisfies  Am=o 
and  which,  together  with  its  differential  quotients  of  the  first  two 
orders,  is  for  all  values  of  x  and  y  within  a  given  area  one-valued  and 
continuous,  and  which  has  for  points  on  the  boundary  of  the  area 
arbitrarily  given  values.-  Riemann  called  this  ''  Dirichlet's  principle," 
but  the  same  theorem  was  stated  by  Green  and  proved  analytically  by 
Sir  William  Thomson.  It  follows  then  that  w  is  uniquely  determined 
for  all  points  within  a  closed  surface,  if  u  is  arbitrarily  given  for  all 
points  on  the  curve,  whilst  v  is  given  for  one  point  within  the  curve. 
In  order  to  treat  the  more  complicated  case  where  w  has  n  values  for 
one  value  of  z,  and  to  observe  the  conditions  about  continuity,  Rie- 
mann invented  the  celebrated  surfaces,  known  as  "Riemann's  sur- 
faces," consisting  of  n  coincident  planes  or  sheets,  such  that  the  pas- 
sage from  one  sheet  to  another  is  made  at  the  branch-points,  and  that 
the  n  sheets  form  together  a  multiply-connected  surface,  which  can 
be  dissected  by  cross-cuts  into  a  singly-connected  surface.  The  n- 
valued  function  w  becomes  thus  a  one-valued  function.  Aided  by 
researches  of  Jacob  Lliroth  (1844-1910)  of  Freiburg  and  of  R.  F.  A. 
Clebsch,  W.  K.  Clifford  brought  Riemann's  surface  for  algebraic  func- 
tions to  a  canonical  form,  in  which  only  the  last  two  of  the  n  leaves 
ire  multiply-connected,  and  then  transformed  the  surface  into  the 

1  Bull.  Am.  Math.  Soc,  Vol.  23,  1916,  p.  S. 

2  0.  Henrici  "Theory  of  Functions,"  Nature,  Vol.  43,  1891,  p.  322. 


THEORY  OF  FUNCTIONS  423 

surface  of  a  solid  with  p  holes.  This  surface  with  p  holes  had  been 
considered  before  Clifford  by  A.  Tonelli,  and  was  probably  used  by 
Riemann  himself.^  A.  Hurwitz  of  Zurich  discussed  the  question,  how 
far  a  Riemann's  surface  is  determinate  by  the  assignment  of  its  number 
of  sheets,  its  branch-points  and  branch-Hnes. 

Riemann's  theory  ascertains  the  criteria  which  will  determine  an 
analytical  function  by  aid  of  its  discontinuities  and  boundary  condi- 
tions, and  thus  defines  a  function  independently  of  a  mathematical 
expression.  In  order  to  show  that  two  different  expressions  are 
identical,  it  is  not  necessary  to  transform  one  into  the  other,  but  it  is 
sufficient  to  prove  the  agreement  to  a  far  less  extent,  merely  in  certain 
critical  points. 

Riemann's  theory,  as  based  on  Dirichlet's  principle  (Thomson's 
theorem),  is  not  free  from  objections  which  have  been  raised  by  L. 
Kronecker,  K.  Wcierstrass,  and  others.  In  consequence  of  this, 
attempts  have  been  made  to  graft  Riemann's  speculations  on  the 
more  strongly  rooted  methods  of  K.  Weierstrass.  The  latter  developed 
a  theory  of  functions  by  starting,  not  with  the  theory  of  potential, 
but  with  analytical  expressions  and  operations.  Both  applied  their 
theories  to  Abelian  functions,  but  there  Riemann's  work  is  more 
general.- 

Following  a  suggestion  found  in  Riemann's  Habilitationsschrift,  H. 
Hankel  prepared  a  tract,  Unendlich  oft  oscilUrende  uiid  unstetige  Funk- 
tionen,  Tubingen,  1S70,  giving  functions  which  admit  of  an  integral, 
but  where  the  existence  of  a  differential  coefficient  remains  doubtful. 
He  supposed  continuous  curves  generated  by  the  motion  of  a  point 
to  and  fro  with  infinitely  numerous  and  infinitely  small  oscillations, 
thus  presenting  ''a  condensation  of  singularities"  at  every  point,  but 
possessing  no  definite  direction  nor  differential  coefficient.  These 
novel  ideas  were  severely  critici  ed,  but  were  finally  cleared  up  by 
K.  Weierstrass'  well-known  rigorous  example  of  a  continuous  curve 
totally  bereft  of  derivatives.  Hermann  Hankel  (1839-1873)  satisfied 
at  Leipzig  the  gymnasium  requirements  in  ancient  languages  by  read- 
ing the  ancient  mathematicians  in  the  original.  He  studied  at  Leipzig 
under  A.  F.  Mobius,  at  Gottingen  under  G.  F.  B.  Riemann,  at  BerUn 
under  K.  Weierstrass  and  L.  Kronecker.  He  became  professor  at 
Erlangen  and  Tubingen.  The  interest  of  his  lectures  was  enhanced 
by  his  emphasis  upon  the  history  of  his  subject.  In  1867  appeared  his 
Thcorie  dcr  Complexen  Zahlensysteme.  His  brilliant  Geschichte  der 
Mathematik  in  Alterthum  und  Mittelaltcr  came  out  in  1S74  as  a  post- 
humous publication. 

Karl  Weierstrass  (1S15-1897)  was  born  in  Ostenfelde,  a  village  in 
Westphalia.  He  attended  a  gymnasium  at  Paderborn  where  he  became 
interested  in  the  geometric  researches  of  J.  Steiner.    He  entered  the 

'  Math.  Annalen.  Vol.  45,  p.  142. 

20.  Henrici,  Nature,  Vol.  43,  1891,  p.  ^2^. 


424  A  HISTORY  OF  MATHE^IATTCS 

University  of  Bonn  as  a  student  in  law  but  all  by  himself  he  studied 
also  mathematics,  particularly  P.  S.  Laplace.  Wilhelm  Diesterweg 
and  J.  Plucker,  who  lectured  in  Bonn,  did  not  influence  him.  Seeing 
in  a  student  note-book  a  transcript  of  Christof  Gudermann's  lectures 
on  elliptic  transcendents,  Weierstrass  went  in  1839  to  Mlinster,  where 
he  was  during  one  semester  the  only  student  to  attend  Gudermann's 
lectures  on  this  topic  and  on  analytical  spherics.  Christof  Guderman:i 
(1798-1851)  whose  researches  on  hyperbolic  functions  led  to  a  func- 
tion ta7t~^  {sink  x),  called  the  "  Gudermannian,"  was  a  favorite  teacher 
of  Weierstrass.  Then  he  became  a  gymnasium  teacher  at  Mlinster, 
then  at  Deutsch-Krone  in  western  Prussia  where  he  taught  science, 
also  gymnastics  and  writing,  and  finally  at  Braunsberg  where  he 
entered  upon  the  study  of  Abelian  functions.  It  is  told  that  he  missed 
one  morning  an  eight-o'clock  class.  The  director  of  the  gymnasium 
went  to  his  room  to  ascertain  the  cause,  and  found  him  working 
zealously  at  a  research  which  he  had  begun  the  evening  before  and 
continued  through  the  night,  being  unconscious  that  morning  had 
come.  He  asked  the  director  to  excuse  his  lack  of  punctuality  to 
his  class,  for  he  hoped  soon  to  surprise  the  world  by  an  important 
discovery.  While  at  Braunsberg  he  received  an  honorary  doctorate 
from  Konigsberg  for  scientific  papers  he  had  published.  In  1855  E.  E. 
Kummer  went  from  Breslau  to  Berlin;  he  expressed  it  as  his  opinion 
that  the  paper  on  Abelian  functions  was  not  sufiicient  guarantee  that 
Weierstrass  was  the  proper  man  to  train  young  mathematicians  at 
Breslau.  So  Ferdinand  Joachimsthal  (1818-1861)  was  appointed 
there,  but  Kummer  secured  for  Weierstrass  in  1856  a  position  at  the 
Gewerbeakademie  in  Berlin  and  at  the  same  time  an  Extraordinariat 
at  the  University.  The  former  he  held  until  1864  when  he  received  an 
Ordinariat  at  the  University  as  successor  to  the  aged  Martin  Ohm. 
In  that  year  E.  E.  Kummer  and  Weierstrass  organized  an  official 
mathematical  seminar,  P.  G.  L.  Dirichlet  having  held  before  this  a 
private  seminar.  It  is  noteworthy  that  Weierstrass  did  not  begin  his 
university  career  as  a  professor  until  his  forty-ninth  year,  a  time  when 
many  scientists  cease  their  creative  work.  K.  Weierstrass,  E.  E. 
Kummer,  and  L.  Kronecker  added  lustre  to  the  University  of  Berlin 
which  previously  had  been  made  famous  by  the  researches  of  P.  G.  L. 
Dirichlet,  J.  Steiner,  and  C.  G.  J.  Jacobi.  Especially  through  Weier- 
strass unprecedented  stress  came  to  be  put  upon  rigor  of  demon- 
stration. The  movement  toward  arithmetization  of  mathematics  re- 
ceived through  Kronecker  and  Weierstrass  its  greatest  emphasis.  The 
number-concept,  especially  that  of  the  positive  integer,  was  to  become 
the  sole  foundation,  and  the  space-concept  was  to  be  rejected  as  a 
primary  concept. 

As  early  as  1849  Weierstrass  began  to  investigate  and  write  on 
Abelian  integrals.  In  1863  and  xS66  he  lectured  on  the  theory  of 
Abelian  functions  and  Abelian  transcendents.    No  authorized  publi- 


THEORY  OF  FUNCTIONS 


425 


cation  of  these  lectures  was  made  in  his  lifetime,  but  they  became 
known  in  part  through  researches  based  upon  them  that  were  written 
by  some  of  his  pupils,  E.  Netto,  F.  Schottky,  Georg  Valentin,  F. 
Kotter,  Georg  Hettner  (1S54-1914),  and  Johannes  Knoblauch  (1855- 
19 1 5).  Hettner  and  Knoblauch  prepared  Weierstrass'  lectures  on  the 
theory  of  Abelian  transcendents  for  the  fourth  volume  of  his  collected 
works.  In  191 5  appeared  the  fifth  volume,  on  elliptic  functions,  edited 
by  Knoblauch.  Weierstrass  had  selected  Hettner  to  edit  the  works  of 
C.  W.  Borchardt  (18S8),  also  the  last  two  volumes  of  Jacobi's  works. 
Knoblauch  lectured  at  the  University  of  Berlin  since  18S9,  his  chief 
field  of  activity  being  differential  geometry.  Another  prominent  pupil 
of  Weierstrass  was  Otto  Stolz  (1S42-1905)  of  the  University  of  Inns- 
bruck. The  difficulty  which  was  exj^erienced  for  many  years  in  as- 
certaining what  were  the  methods  and  results  of  Weierstrass,  was 
set  forth  by  Adolf  Mayer  (1S39-1908)  of  Leipzig  who  at  one  time  had 
put  at  his  disposal  the  manuscript  notes  of  a  lecture  for  only  twenty- 
four  hours.  Mayer  worked  on  differential  equations,  the  calculus  of 
variations  and  mechanics. 

In  1 86 1  Weierstrass  made  the  extraordinary  discovery  of  a  function 
which  is  continuous  over  an  interval  and  does  not  possess  a  derivative 
at  any  point  on  this  interval.  The  function  was  published  by  P.  du 
Bois  Reymond  in  Crelle's  Journal,  Vol.  79,  1874,  p.  29.  In  1835  N.  I. 
Lobachevski  had  sho\^^l  in  a  memoir  the  necessity  of  distinguishing 
between  continuity  and  differentiability.^  Nevertheless,  the  mathe- 
matical world  received  a  great  shock  when  Weierstrass  brought  forth 
that  discovery,  "and  H.  Hankel  and  G.  Cantor  by  means  of  their 
principle  of  condensation  of  singularities  could  construct  analytical 
expressions  for  functions  having  in  any  interval,  however  small,  an 
infinity  of  points  of  oscillation,  an  infinity  of  points  in  which  the  dif- 
ferential coefficient  is  altogether  indeterminate,  or  an  infinity  of  points 
of  discontinuity"  (J.  Pierpont).  J.  G.  Darboux  gave  new  examples  of 
continuous  functions  having  no  derivatives.  Formerly  it  had  been 
generally  assumed  that  every  function  had  a  derivative.  A.  M.  Am- 
pere was  the  first  who  attempted  to  prove  analytically  (1806)  the 
existence  of  a  derivative,  but  the  demonstration  is  not  valid.  In 
treating  of  discontinuous  functions,  J.  G.  Darboux  established  rigor- 
ously the  necessary  and  sufficient  condition  that  a  continuous  or  dis- 
continuous function  be  susceptible  o^  integration.  He  gave  fresh 
evidence  of  the  care  that  must  be  exercised  in  the  use  of  series  by  giv- 
ing an  example  of  a  series  always  convergent  and  continuous,  such 
that  the  series  formed  by  the  integrals  of  the  terms  is  always  con- 
vergent, and  yet  does  not  represent  the  integral  of  the  first  series.^ 

Central  in  Weierstrass'  view-point  is  the  concept  of  the  "analytic 
function."    The  name,  "general  theory  of  analytic  functions,"  says 

*  G.  B.  Halsted's  transl.  of  A.  Vasiliev's  Address  on  Lobachevski,  p.  23. 

*  Notice  sur  l^s  Travaux  Scienlijiques  de  M.  Gaston  Darboux,  Paris,  1884. 


426  A  HISTORY  OF  MATHEMATICS 

A.  Hurwitz,^  applies  to  two  theories,  that  of  A.  L.  Cauchy  and  G.  F.  B. 
Riemann,  and  that  of  K.  Weierstrass.  The  two  emanate  from  dif- 
ferent definitions  of  a  function.  J.  Lagrange,  in  his  Theorie  des  fonc- 
tions  analytiques  had  tried  to  prove  the  incorrect  theorem  that  every 
continuous  (stetige)  function  can  be  expanded  in  a  power  series.  K. 
Weierstrass  called  every  function  "analytic"  when  it  can  be  expanded 
into  a  power  series,  which  is  the  centre  of  Weierstrass'  theory  of  ana- 
lytic functions.  All  properties  of  the  function  are  contained  in  mice  in 
the  power  series,  with  its  coefficients  ci,  C2,  .  .  ,  Cm,  .  .  The  beha\dor 
of  a  power  series  on  the  circle  of  convergence  C  had  received  considera- 
tion long  before  this  time.  N.  H.  Abel  had  demonstrated  that  the 
power  series  having  a  determinate  value  in  a  point  on  the  circle  of 
convergence  C  tends  uniformly  tOAvard  that  value  when  the  variable 
approaches  that  point  along  a  path  which  does  not  touch  the  circle. 
If  two  power  series  involve  a  complex  variable,  whose  circles  of 
convergence  overlap,  so  that  the  two  series  have  the  same  value  for 
every  point  common  to  the  too  circular  areas,  then  Weierstrass  calls 
each  power  series  a  direct  continuation  of  the  other.  Using  several 
such  series,  K.  Weierstrass  introduces  the  idea  of  a  monogenic  system 
of  power  series  and  then  gives  a  more  general  definition  of  analytic 
function  as  a  function  which  can  be  defined  by  a  monogenic  system 
of  power  series.  In  1S72  the  Frenchman  Ch.  Meray  gave  independ- 
ently a  similar  definition.  In  case  of  a  uniform  (eindeutige)  function, 
the  points  in  a  complex  plane  are  either  within  the  circle  of  conver- 
gence of  the  power  series  in  the  system  or  else  they  are  without.  The 
totality  of  the  former  points  constitutes  the  "field  of  continuity" 
(Stetigkeitsbereich)  of  the  function.  This  field  constitutes  an  ag- 
gregate of  "inner"  points  that  is  dense;  if  this  continuum  is  given, 
then  there  exist  always  single-valued  analytic  functions  possessing 
this  field  of  continuity,  as  was  first  proved  by  G.  M.  Mittag-Leffler, 
later  by  C.  Runge  and  P.  Stackel.  The  points  on  the  boundary  of 
this  field,  called  "singular  points,"  constitute  by  themselves  a  set 
of  points,  by  the  properties  of  which  K.  Weierstrass  classifies  the 
function  (1876).  This  classification  was  studied  also  by  C.  Guichard 
(1883)  and  by  G.  M.  Mittag-Lefller,  making  use  of  theorems  on  point 
sets,  as  developed  in  1879-1885  by  G.  Cantor  and  by  I.  O.  Bendixson 
and  E.  Phragmen,  both  of  Stockholm.  Thus,  transfinite  numbers 
began  to  play  a  part  in  the  theory  of  functions.  Single- valued  analytic 
functions  resolve  themselves  into  two  classes,  the  one  class  in  which 
the  singular  points  form  an  enumerable  (abzahlbares)  aggregate,  the 
other  class  in  which  they  do  not. 

Abel  had  proposed  the  problem,  if  one  supposes  the  power  series 
convergent  for  all  positive  values  less   than  r,   find   the  limit  to 
which  the  function  tends  when  x  approaches  r.     The  first  sub- 
stantial advance  to  a  solution  of  Abel's  problem  was  made  in  1880 
'  A.  Hurwitz  in  Verh.  des  i.  Intern.  Congr.,  Zurich,  iSgy,  Leipzig,  1898,  pp.  91-112. 


THEORY  OF  FUXC  TIOXS  427 

by  G.  Frobenius  and  in  1882  by  O.  Holder,  but  neither  of  them 
developed  conditions  that  are  both  necessary  and  sufficient  for 
the  establishment  of  the  convergence  of  their  expressions.  Finally 
in  1892  J.  Hadamard  obtained  ex])ressions  which  include  those  of 
G.  Frobenius  and  0.  Holder  and  determined  the  conditions  under 
which  they  converge  on  the  circle  of  convergence.  The  problem  pre- 
sented itself  now  thus:  To  set  up  analytic  expressions  of  the  complex 
variable  x  that  are  linear  in  the  constants  c„  and  also  represent  the 
function  given  by  the  power  series,  or  rather  a  branch  of  this  function 
in  a  field  D,  in  such  a  manner  that  they  converge  uniformly  in  the  in- 
terior of  D  and  diverge  in  the  exterior.  The  first  important  step 
toward  the  resolution  of  this  matter  was  taken  in  1895  by  E.  Borel 
who  proved  that  the  expression 

Urn    2  (r<,+ci.v+  .  .  +cy)e-".-—— 

converges  not  only  in  all  regular  points  (points  reguliers)  of  the  circle 
of  convergence  of  the  power  series,  but  even  beyond  that,  within  a 
summation  polygon.  E.  Borel  held  the  view  that  his  formula  gave  the 
sum  of  the  power  series  even  for  points  where  it  diverges.  This  inter- 
pretation of  Borel's  results  was  resisted  by  Gosta  Magmis  Mittag-Leffier 
(1846-  )  of  Stockholm,  the  founder^  of  the  journal  Acta  mathe- 
matica,  and  of  a  "Mathematical  Institute"  (in  1916)  to  further  mathe- 
matical research  in  the  Scandinavian  countries.  Mittag-Leffler  con- 
ducted important  researches  along  the  above  line.  E.  Borel's  statement 
implies  that  his  formula  extends  the  boundaries  of  the  theory  of  ana- 
lytic functions  beyond  the  classic  region,  which  is  denied  by  Mittag- 
Lefiler.  The  latter  published  in  1898  studies  on  a  problem  more  gen- 
eral than  that  of  Borel.  If  a  ray  ap  revolves  about  a  through  an  angle 
2  7r,  the  variable  distance  ap  always  exceeding  a  fixed  value  /,  a  sur- 
face is  generated  which  Mittag-Leffler  calls  a  star  (Stern)  with  the 
center  a.  A  star  E  is  called  a  convergence  star  (Konvergenzstem)  for  a 
definite  arithmetical  expression,  if  the  latter  converges  uniformly  for 
each  region  within  E,  but  diverges  for  every  outside  point.  He  shows 
that  to  each  analytic  function  there  corresponds  a  principal  star,  and 
that  there  is  an  infinite  number  of  arithmetical  expressions  for  a  given 
star.  Equivalent  results  were  obtained  by  C.  Runge.  E.  Borel  gave 
in  1912  an  example  of  an  analytic  function  which,  by  an  extension  of 
the  concept  of  a  derivative  so  as  to  pass  to  the  limit  not  through  all 
the  neighboring  points  but  only  through  those  belonging  to  certain 
dense  aggregates,  has  a  certain  linear  continuation  beyond  the  do- 
main of  existence.  Studies  of  monogenic  uniform  functions  along 
the  line  of  E.  Borel  and  G.  M.  Mittag-Leffler  have  been  made  also 
by  G.  Vivanti,  Marcel  Riesz,  Ivar  Fredholm,  and  E.  Phragmen. 

^  See  G.  M.  Mittag-Leffler  in  A  Hi  del  TV  Congr.  Litem,  Roma,  1908.    Roma,  1909, 
Vol.  I,  p.  69.    H^'.re  Mittag-Leffler  gives  a  summary  of  recent  results. 


428  A  HISTORY  OF  MATHEMATICS 

Interesting  is  the  manner  in  which  K.  Weierstrass  in  Berlin  and 
G.  F.  B.  Riemann  in  Gottingen  influenced  each  other.  We  have 
seen  that  Weierstrass  defined  functions  of  a  complex  variable  by  the 
power  series  and  avoids  geometrical  means.  Riemann  begins  with 
certain  differential  equations  in  the  region  of  mathematical  physics. 
In  1856  Riemann  was  urged  by  his  friends  to  publish  a  resume  of 
his  researches  on  Abelian  functions,  "be  it  ever  so  crude,"  because 
Weierstrass  was  at  work  on  the  same  subject.  Riemann's  publication 
induced  Weierstrass  to  withdraw  from  the  press  a  memoir  he  had 
presented  to  the  Berlin  Academy  in  1857,  because,  as  he  himself  says, 
"Riemann  published  a  memoir  on  the  same  problem  which  rested  on 
entirely  different  foundations  from  mine,  and  did  not  immediately 
reveal  that  in  its  results  it  agreed  completely  with  my  own.  The 
proof  of  this  required  investigations  which  were  not  quite  easy,  and 
took  much  time;  after  this  difficulty  had  been  removed  a  radical 
remodelling  of  my  dissertation  seemed  necessary."  In  1875  Weier- 
strass wrote  H.  A.  Schwarz:  "The  more  I  ponder  over  the  principles 
of  the  theory  of  functions — and  I  do  so  incessantly — the  stronger 
grows  my  conviction  that  it  must  be  built  up  on  the  foundation  of 
algebraical  truths,  and  that,  therefore,  to  employ  for  the  truth  of 
simple  and  fundamental  algebraical  theorems  the  'transcendental,' 
if  I  may  say  so,  is  not  the  correct  way,  however  enticing  prima  vista 
the  considerations  may  be  by  which  Riemann  has  discovered  many 
of  the  most  important  properties  of  algebraical  functions."  This 
refers  mainly  to  the  "Thomson-Dirichlet  Principle,"  the  validity  of 
which  depended  on  a  certain  minimum  theorem  which  was  shown 
by  Weierstrass  to  rest  upon  unsound  argument. 

It  has  been  objected  that  K.  Weierstrass'  definition  of  analytic 
functions  is  based  on  power  series.  A.  L.  Cauchy's  definition,  which 
was  adopted  by  G.  F.  B.  Riemann,  is  not  open  to  this  objection,  but 
labors  under  the  burden  of  requiring  at  the  start  the  most  difficult 
forms  of  the  theory  of  limits.  According  to  A.  L.  Cauchy  a  function 
is  analytic  (his  "synectic")  if  it  possesses  a  single- valued  differential 
coefficient.  Using  Cauchy's  integral  theorem  (Integralsatz),  it  follows 
that  the  synectic  function  admits  not  only  of  a  single-valued  differen- 
tiation but  also  of  a  single-valued  integration.  Giacinto  Morera 
(1856-1909)  of  Turin  showed  that  the  synectic  function  might  be 
defined  by  the  single-valued  integration.  More  recent  researches, 
1883-1895,  which  aim  at  a  rigorous  exposition  of  A.  L.  Cauchy's 
integral  theorem,  are  due  to  M.  Falk,  E.  Goursat,  M.  Lerch,  C.  Jor- 
dan, and  A.  Pringsheim.  Cauchy's  theorem  may  be  stated :  If  the  func- 
tion /(z)  is  synectic  in  a  continuum  in  which  every  simply  closed 

curve  forms  the  boundary  of  an  area,  then  the  integral    lj{z)dz  is 

always  zero,  if  it  is  extended  over  a  closed  curve  which  lies  wholly 


THEORY  OF  FUNCTIONS  429 

within  the  continuum.  Here  the  questions  arise,  what  is  a  curve,  a 
closed  curve,  a  simply  closed  curve? 

Analytic  functions  of  several  variables  were  treated  by  C.  G.  J. 
Jacobi  in  1832,  in  his  Consider ationes  generates  de  trans cendentihis 
Abelianis,  but  received  no  attention  until  Weierstrass  set  himself  the 
task  presented  to  him  by  the  study  of  Abelian  functions,  to  find  a 
solid  foundation  for  functions  of  several  variables  that  would  corre- 
spond to  his  treatment  of  functions  of  one  variable.  He  obtained  a 
fundamental  theorem  on  null-places;  he  also  enunciated,  without 
proof,  the  theorem  that  each  single-valued  (eindeutig)  and  in  a  finite 
region  meromorphic  function  of  several  variables  can  be  represented 
as  the  quotient  of  two  integral  functions,  i.  e.  of  two  bestandig  con- 
vergent power  series.  This  theorem  was  proved  in  1883  by  H.  Poin- 
care  for  two  variables  and  in  1895  by  Pierre  Cousin  of  Bordeaux  for 
n  variables.  Later  researches  are  by  H.  Hahn  (1905),  P.  Boutroux 
(1905),  G.  Faber  (1905),  and  F.  Hartogs  (1907). 

Dirichlet's  principle  has  repeatedly  commanded  attention.  The 
question  of  its  rigor  has  been  put  by  E.  Picard  as  follows:  ^  "The 
conditions  at  the  limits  that  one  is  led  to  assume  are  very  different 
according  as  it  is  question  of  an  equation  of  which  the  integrals  are 
or  are  not  analytic.  A  {.y^e.  of  the  first  case  is  given  by  the  problem 
generalized  by  P.  G.  L.  Dirichlet;  conditions  of  continuity  there  play 
an  essential  part,  and,  in  general,  the  solution  cannot  be  prolonged 
from  the  two  sides  of  the  continuum  which  serves  as  support  to  the 
data;  it  is  no  longer  the  same  in  the  second  case,  where  the  disposition 
of  this  support  in  relation  to  the  characteristics  plays  the  principal 
role,  and  where  the  field  of  existence  of  the  solution  presents  itself 
under  wholly  different  conditions.  .  .  .  From  antiquity  has  been 
felt  the  confused  sentiment  of  a  certain  economy  in  natural  phenomena; 
one  of  the  first  precise  examples  is  furnished  by  Fermat's  principle 
relative  to  the  economy  of  time  in  the  transmission  of  light.  Then  we 
came  to  recognize  that  the  general  equations  of  mechanics  correspond 
to  a  problem  of  minimum,  or  more  exactly  of  variation,  and  thus  we 
obtained  the  principle  of  virtual  velocities,  then  Hamilton's  principle, 
and  that  of  least  action.  A  great  number  of  problems  appeared  then 
as  corresponding  to  minima  of  certain  definite  integrals.  This  was  a 
very  important  advance,  because  the  existence  of  a  minimum  could 
in  many  cases  be  regarded  as  evident,  and  consequently  the  demon- 
stration of  the  existence  of  the  solution  was  effected.  This  reasoning 
has  rendered  immense  services;  the  greatest  geometers,  K.  F.  Gauss 
in  the  problem  of  the  distribution  of  an  attracting  mass  corresponding 
to  a  given  potential,  G.  F.  B.  Riemann  in  his  theory  of  Abelian  func- 
tions, have  been  satisfied  with  it.  To-day  our  attention  has  been 
called  to  the  dangers  of  this  sort  of  demonstration;  it  is  possible  for 
the  minima  to  be  simply  limits  and  not  to  be  actually  attained  by 
'  Congress  of  Arts  and  Science,  St.  Louis,  1904,  \'ol.  I.  p.  510. 


430  A  HISTORY  OF  MATHEMATICS 

veritable  functions  possessing  the  necessary  properties  of  continuity. 
We  are,  therefore,  no  longer  content  with  the  probabilities  offered 
by  the  reasoning  long  classic." 

David  Hilbert  in  1899  spoke  as  follows:  ^  "Dirichlet's  principle 
owed  its  celebrity  to  the  attractive  simplicity  of  its  fundamental 
mathematical  idea,  to  the  undeniable  richness  of  its  possible  apphca- 
tions  in  pure  and  applied  mathematics  and  to  its  inherent  persuasive 
power.  But  after  Weierstrass'  criticism  of  it,  Dirichlet's  principle 
was  considered  as  only  of  historical  interest  and  discarded  as  a  means 
of  solving  the  boundary-value  problem.  C.  Neumann  deplores  that 
this  beautiful  principle  of  Dirichlet,  formerly  used  so  much,  has  no 
doubt  passed  away  forever.  Only  A.  Brill  and  M.  Noether  arouse 
new  hopes  in  us  by  giving  expression  to  the  conviction  that  Dirichlet's 
principle,  being  so  to  speak  an  imitation  of  nature,  may  sometime 
receive  new  life  in  modihed  form."  Hilbert  proceeds  thereupon  to 
rehabilitate  the  Principle,  which  involves  a  special  problem  in  the  cal- 
culus of  variations.  Dirichlet's  procedure  was  briefly  thus:  On  the 
xy  plane  erect  at  the  points  of  the  boundary  curve  perpendiculars  the 
lengths  of  which  represent  the  boundary  values.  Among  the  surfaces 
z=f{x,y)  which  are  bounded  by  the  space-curve  thus  obtained,  select  the 

one  for  which  the  value  of  the  integral  /(/)  =1    /'()+(^")    f  dxdy 

is  a  minimum.  As  shown  by  the  calculus  of  variations,  that  surface 
is  necessarily  a  potential  surface.  By  reference  to  such  a  procedure 
G.  F.  B.  Riemann  thought  he  had  settled  the  existence  of  the  solution 
of  boundary-value  problems.  But  K.  Weierstrass  made  it  plain  that 
among  an  infinite  number  of  values  there  does  not  necessarily  exist 
a  minimum  value;  a  minimum  surface  may  therefore  not  exist.  D. 
Hilbert  generalizes  Dirichlet's  principle  in  this  manner:  "Every  prob- 
lem of  the  calculus  of  variations  has  a  solution,  as  soon  as  restricting 
assumptions  suitable  to  the  nature  of  the  given  boundary  conditions 
are  satisfied  and,  if  necessary,  the  concept  of  the  solution  receives  a 
fitting  extension."  D.  Hilbert  shows  how  this  may  be  used  in  finding 
rigorous,  yet  simple,  existence  proofs.  In  1901  it  was  used  in  disserta- 
tions prepared  by  E.  R.  Hedrick  and  C.  A.  Noble. 

Taking  a  birds'  eye  view  of  the  development  of  the  theory  of  func- 
tions during  the  nineteenth  century  since  the  time  of  A,  L.  Cauchy, 
James  Pierpont  said  in  1904:-  "Weierstrass  and  Riemann  develop 
Cauchy's  theory  along  two  distinct  and  original  paths.  Weierstras? 
starts  with  an  explicit  analytic  expression,  a  power  series,  and  define? 
his  function  as  the  totality  of  its  analytical  continuations.  No  appeal 
is  made  to  geometric  intuition,  his  entire  theory  is  strictly  arithmetical. 
Riemann  growing  up  under  Gauss  and  Dirichlet,  not  only  relies  largely 

^  Jahresh.  d.  d.  Math.  Vereinig.,  Vol.  8,  1900,  p.  185. 
2  Bull.  Am.  Math.  Soc,  2.  S.,  Vol.  11,  1904,  p.  137. 


THEORY  OF  FUNCTIONS  431 

on  geometric  intuition,  but  also  does  not  hesitate  to  impress  mathe- 
matical physics  into  his  service.  Two  noteworthy  features  of  his 
theory  are  the  many-leaved  surfaces  named  after  him,  "and  the  ex- 
tensive use  of  conformal  representation.  The  history  of  functions 
as  first  developed  is  largely  a  theory  of  algebraic  functions  and  their 
integrals.  A  general  theory  of  functions  is  only  slowly  evolved.  For 
a  long  time  the  methods  of  Cauchy,  Riemann,  and  Weierstrass  were 
cultivated  along  distinct  lines  by  their  respective  pupils.  The  schools 
of  Cauchy  and  Riemann  were  first  to  coalesce.  The  entire  rigor 
which  has  recently  been  imparted  to  their  methods  has  removed  all 
reason  for  founding,  as  Weierstrass  and  his  school  have  urged,  the 
theory  of  functions  on  a  single  algorithm,  viz.,  the  power  series.  We 
may  therefore  say  that  at  the  close  of  the  century  there  is  only  one 
theory  of  functions,  in  which  the  ideas  of  its  three  great  creators  are 
harmoniously  united." 

The  study  of  existence  theorems,  particularly  in  the  theory  of  alge- 
braic functions  and  the  calculus  of  variations,  began  with  Cauchy. 
For  implicit  functions  he  assumed  that  they  were  expressible  as  power 
series,  a  restriction  removed  by  U.  Dini  of  Pisa.  SimpHfications  are 
due  to  R.  Lipschitz  of  Bonn.  Existence  theorems  of  sets  of  implicit 
functions  were  studied  by  G.  A.  Bliss  of  Chicago  in  the  Princeton 
Colloquium  of  1909.  By  means  of  a  sheet  of  points  Bhss  deduces 
from  an  initial  solution  at  an  ordinary  point  a  sheet  of  solutions 
somewhat  analogous  to  K.  Weierstrass'  analytical  continuation  of  a 
branch  of  a  curve. 

Accompanying  and  immediately  following  Riemann's  time  there 
was  a  development  of  the  theory  of  algebraic  functions,  that  was 
partly  geometric  in  character  and  not  purely  along  the  line  of  function 
theory.  A.  Brill  and  M.  Noether  '  in  1894  marked  five  directions  of 
advance:  First,  the  geometrico-algebraic  direction  taken  by  G.  F.  B. 
Riemann  and  G.  Roch  in  the  years  1862-1866,  then  by  R.  F.  A. 
Clebsch  1863  to  1865,  by  Clebsch  and  P.  Gordan  since  1865  and  since 
187 1  by  A.  Brill  and  M.  Noether;  second,  the  algebraic  direction, 
followed  by  L.  Kronecker  and  K.  Weierstrass  since  i860,  more  gen- 
erally known  since  1872,  and  in  1880  taken  up  by  E.  B.  Christoffel; 
third,  the  invariantal  direction,  represented  since  1877  by  H.  Weber, 
M.  Noether,  E.  B.  Christoffel,  F.  Klein,  F.  G.  Frobenius,  and  F. 
Schottky;  Fourth,  the  arithmetical  direction  of  R.  Dedekind  and  H. 
Weber  since  1880,  of  L.  Kronecker  since  1881,  of  K.  W.  S.  Hensel  and 
others;  Fifth,  the  geometrical  direction  taken  by  C.  Segre  and  G. 
Castelnuovo  since  18S8. 

Hermann  Amandus  Schwarz  (1S45-        )  of  Berlin  a  pupil  of  K. 

Weierstrass,   has  given   the  conform   representation   {Abbildung)   of 

various  surfaces  on  a  circle.    G.  F.  B.  Riemann  had  given  a  general 

theorem  on  the  conformation  of  a  given  curve  with  another  curve. 

1  A.  Brill  and  M.  Noelher,  Jahrb.  d.  d.  Malh.  Vercinivunp,  Vol.  3  "p.  2S7. 


432  A  HISTORY  OF  MATHEMATICS 

In  transforming  by  aid  of  certain  substitutions  a  polygon  bounded 
by  circular  arcs  into  another  also  bounded  by  circular  arcs,  Schwarz 
was  led  to  a  remarkable  differential  equation  ip(u',  t)  =  ^{u,  t),  where 
\J/(u,  t)  is  the  expression  which  Cayley  called  the  "  Schwarzian  deriva- 
tive," and  which  led  J.  J.  Sylvester  to  the  theory  of  reciprocants. 
Schwarz's  developments  on  minimum  surfaces,  his  work  on  hyper- 
geometric  series,  his  inquiries  on  the  existence  of  solutions  to  important 
partial  differential  equations  under  prescribed  conditions,  have  se- 
cured a  prominent  place  in  mathematical  literature. 

Modular  functions  were  at  first  considered  merely  as  a  by-product 
of  elliptic  functions,  growing  out  of  the  study  of  transformations. 
After  the  epoch  making  creations  of  E.  Galois  and  G.  F.  B.  Riemann, 
the  subject  of  elliptic  modular  functions  was  developed  into  an  in- 
dependent theory,  mainly  by  the  efforts  of  H.  Poincare  and  F.  Klein, 
which  stands  in  close  relation  to  the  theory  of  numbers|  algebra  and 
synthetic  geometry.  F.  Klein  began  to  lecture  on  this  subject  in 
1877;  researches  bearing  upon  this  were  pursued  also  by  his  then 
pupils  W.  Dyck,  Joseph  Gierster,  and  A.  Hurwitz.  One  of  the  problems 
of  modular  functions  is,  to  determine  all  subgroups  of  the  linear  group 
a-^  =  (ax+i3)  /(7.V+8),  where  a,  /?,  7,  ^  are  integers  and  ah- ^y 7^0. 
F.  Klein's  Vorlesungen  uber  das  Ikosader,  Leipzig,  1884,  is  a  work 
along  this  hne.  As  an  extended  continuation  of  that  are  F.  Klein's 
Vorlesungen  uher  die  Theorie  der  cUiptischen  M odulfunctionen,  gotten 
out  by  Robert  Fricke  (Vol.  i,  1890,  Vol.  II,  1892)  and  as  a  still 
further  generalization  we  have  the  theory  of  the  general  linear  auto- 
morphic  functions,  developed  mainly  by  F.  Klein  and  H.  Poincare. 
In  1897,  under  the  joint  authorship  of  Robert  Fricke  and  Felix  Klein, 
there  appeared  the  first  volume  of  the  Vorlesungen  uber  die  Theorie 
der  Automorphen  Fundionen,  the  second  volume  of  which  did  not  ap- 
pear until  1912,  after  the  theory  had  come  under  the  influence  of  the 
critical  tendencies  due  to  K.  Weierstrass  and  G.  Cantor,  and  after 
E.  Picard  and  H.  Poincare  had  brought  out  further  incisive  researches. 
It  has  been  noted  that  F.  Klein's  own  publications  on  these  topics 
are  in  the  order  in  which  the  subject  itself  sprang  into  existence. 
"Historically,  the  theory  of  automorphic  functions  developed  from 
that  of  the  regular  solids  and  modular  functions.  At  least  this  is  the 
path  which  F.  Klein  followed  under  the  influence  of  the  well-known 
researches  of  Schwarz  and  of  the  early  publications  of  H.  Poincare. 
If  H.  Poincare  brings  in  also  other  considerations,  namely  the  arith- 
metic methods  of  Ch.  Hermite  .  .  .  and  the  function-theoretical 
problems  of  Fuchs  with  regard  to  single  valued  inversion  of  the  solu- 
tions of  linear  differential  equations  of  the  second  order  (eindeutige 
Umkehr  der  Losungen  .  .  .  ),  these  topics  in  turn  go  back  to  the  very 
regions  of  thought  from  which  have  grown  the  theories  of  the  regular 
solids  and  the  elliptic  modular  functions."  H.  Poincare  published 
on  this  subject  in  Math.  Annalen,  Vol.  19,  "  Sur  les  fonctions  uniformes 


THEORY  OF  FUNCTIONS  4,33 

qui  se  reproduisent  par  des  substitutions  lineaires,"  in  the  Acta  ma- 
ihematica,  Vol.  i,  a  "Memoire  sur  les  fonctions  fuchsiennes,"  and  a 
procession  of  other  papers  extending  over  many  years.  Recently 
active  along  this  same  line  were  P.  Koebe  and  L.  E.  J.  Brouwer. 

The  question  what  automorphic  forms  can  be  expressed  analytically 
by  the  H.  Poincare  series  has  been  investigated  by  Poincare  himself 
and  also  by  E.  Ritter  and  R.  Fricke  (1901). 

After  the  creation  of  the  theory  of  automorphic  functions  of  a  single 
variable,  mainly  by  F.  Klein  and  H.  Poincare,  similar  generalizations 
were  sought  for  functions  of  several  complex  variables.  The  pioneer 
in  this  field  was  E.  Picard;  other  workers  are  T.  Levi-Civita,  G.  A. 
Bliss,  W.  D.  MacMillan,  and  W.  F.  Osgood  who  lectured  thereon  at  the 
Madison  (Wisconsin)  Colloquium  in  1913.  Charles  Emile  Picard 
C1856 —  )  whose  extensive  researches  on  analysis  have  been  men- 
tioned repeatedly  and  whose  Traite  d' Analyse  is  well  known,  was 
born  in  Paris.  He  studied  at  the  Ecole  Normale  where  he  was  in- 
spired by  J.  G.  Darboux.  In  188 1  he  married  a  daughter  of  Hermite. 
Picard  taught  for  a  shorty  time  at  Toulouse.  Since  1881  he  has  been 
professor  in  Paris  at  the  Ecole  Normale  and  the  Sorbonne. 

Uniformization 

The  uniformization  of  an  algebraic  or  anah^tic  curve,  that  is,  the 
determination  of  such  auxiliary  variables  which  taken  as  independent 
variables  render  the  co-ordinates  of  the  points  of  the  curve  single- 
valued  (eindev.tig)  analytic  functions,  is  organically  connected  with 
the  theory  of  automorphic  functions.  It  was  F.  Klein  and  H.  Poin- 
care who  soon  after  1880  developed  the  theory  of  automorphic  func- 
tions and  introduced  systematically  the  idea  of  the  uniformization  of 
algebraic  curves  which  G.  F.  B.  Riemann  had  visualized  upon  the 
surfaces  named  after  him.  More  recent  researches  on  uniformi  ation 
connect  chiefly  with  the  work  of  H.  Poincare  and  are  due  to  D.  Hil- 
bert  (iQOo),  W.  F.  Osgood,  T.  Broden,  and  A.  M.  Johanson.  In  1907 
followed  important  generalizations  by  H.  Poincare  and  by  P.  Koebe 
of  Leipzig.^  Dirichlet's  Principle,  having  been  estabhshed  upon  a 
s.)und  foundation  by  D.  Hilbert  in  1901,  was  used  as  a  starting  point, 
for  the  derivation  of  new  proofs  of  the  general  principle  of  uniformiza- 
tion, by  P.  Koebe  of  Leipzig  and  R.  Courant  of  Giittingen. 

Important  works  on  the  theory  of  functions  are  the  Coiirs  de  Ch. 
Ilcrmitc,  J .  Tannery's  Thcorie  dcs  Fonctions  d\ine  variable  seule,  A 
Treatise  on  the  Theory  of  Functions  by  James  Harkness  and  Frank  Mor- 
ley,  and  Theory  of  Functions  of  a  Complex  Variable  by  A.  R.  Forsyth. 
A  broad  and  comprehensive  treatise  is  the  Lchrbuch  der  Funktionen- 
theorie  by  W.  F.  Osgood  of  Harvard  University,  the  first  :idition  of 
rt'hich  appeared  in  1907  and  the  second  enlarged  edition  in  191 2. 
'  P.  Koebe.  Alii  del  IV  Congr.,  Rjind,  igoS,  Rci.na,  lyoj,  Vol.  1 1,  p.  2^. 


434  A  HISTORY  OF  MATHEMATICS 

Theory  of  Numbers 

"  Mathematics,  the  queen  of  the  sciences,  and  arithmetic,  the  queen 
of  mathematics."  Such  was  the  dictum  of  K.  F.  Gauss,  who  was 
destined  to  revolutionize  the  theory  of  numbers.  When  asked  who 
was  the  greatest  mathematician  in  Germany,  P.  S.  Laplace  answered, 
Pfaflf.  When  the  questioner  said  he  should  have  thought  Gauss  was, 
Laplace  replied,  "Pfaff  is  by  far  the  greatest  mathematician  in  Ger- 
many; but  Gauss  is  the  greatest  in  all  Europe."  ^  Gauss  is  one  of 
the  three  greatest  masters  of  analysis, — J.  Lagrange,  P.  S.  Laplace,  K. 
F.  Gauss.  Of  these  three  contemporaries  he  was  the  youngest.  While 
the  first  two  belong  to  the  period  in  mathematical  history  preceding 
the  one  now  under  consideration,  Gauss  is  the  one  whose  writings  may 
truly  be  said  to  mark  the  beginning  of  our  own  epoch.  In  him  that 
abundant  fertihty  of  invention,  displayed  by  mathematicians  of  the 
preceding  period,  is  combined  with  rigor  in  demonstration  which  is  too 
often  wanting  in  their  writings,  and  which  the  ancient  Greeks  might 
have  envied.  Unlike  P.  S.  Laplace,  Gauss  strove  in  his  writings  after 
perfection  of  form.  He  rivals  J.  Lagrange  in  elegance,  and  surpasses 
this  great  Frenchman  in  rigor.  Wonderful  was  his  richness  of  ideas; 
one  thought  followed  another  so  quickly  that  he  had  hardly  time  to 
write  down  even  the  most  meagre  outline.  At  the  age  of  twenty 
Gauss  had  overturned  old  theories  and  old  methods  in  all  branches  of 
higher  mathematics;  but  little  pains  did  he  take  to  publish  his  results, 
and  thereby  to  estabUsh  his  priority.  He  was  the  first  to  observe 
rigor  in  the  treatment  of  infinite  series,  the  first  to  fully  recognize 
and  emphasize  the  importance,  and  to  make  systematic  use  of  de- 
terminants and  of  imaginaries,  the  first  to  arrive  at  the  method  of 
least  squares,  the  first  to  observe  the  double  periodicity  of  elliptic 
functions.  He  invented  the  heliotrope  and,  together  with  W.  Weber, 
the  bifilar  magnetometer  and  the  declination  instrument.  He  re- 
constructed the  whole  of  magnetic  science. 

Karl  Friedrich  Gauss  -  (1777-1855),  the  son  of  a  bricklayer,  was 
born  at  Brunswick.  He  used  to  say,  jokingly,  that  he  could  reckon 
before  he  could  talk.  The  marvellous  aptitude  for  calculation  of  the 
young  boy  attracted  the  attention  of  Johann  Martin  Bartels  (1769- 
1836),  afterwards  professor  of  mathematics  at  Dorpat,  who  brought 
him  under  the  notice  of  Charles  William,  Duke  of  Brunswick.  The 
duke  undertook  to  educate  the  boy,  and  sent  him  to  the  Collegium 
Carolinum.  His  progress  in  languages  there  was  quite  equal  to  that 
in  mathematics.  In  1795  he  went  to  Gottingen,  as  yet  undecided 
whether  to  pursue  philology  or  mathematics.  Abraham  Gotthelf 
Kastner  (1719-1800),  then  professor  of  mathematics  there,  and  now 
chiefly  remembered  for  his  Geschichte  der  Mathematik  (1796),  was  not 

1  R.  Tucker,  "Carl  Friedrich  Gauss,"  Nature,  Vol.  15,  1877,  p.  534. 

2  W.  Sartopus  Waltershausen,  Gauss,  ziim  Geddchtniss,  Leipzig,  1856. 


IHEOKV  OF  NUxMBERS  4,:i5 

1  teacher  who  could  inspire  Gauss,  though  Kastner's  German  con- 
temporaries ranked  him  high  and  admired  his  mathematical  and 
poetical  ability.  Gauss  declared  that  Kastner  was  the  first  mathe- 
matician among  the  poets  and  the  first  poet  among  the  mathemati- 
cians. When  not  quite  nineteen  years  old  Gauss  began  jotting  down 
in  a  copy-book  very  brief  Latin  memoranda  of  his  mathematical  dis- 
coveries. This  diary  was  published  in  iqoi.'  Of  the  146  entries,  the 
first  is  dated  March  30,  1796,  and  refers  to  his  discovery  of  a  method 
of  inscribing  in  a  circle  a  regular  polygon  of  seventeen  sides.  This  dis- 
covery encouraged  him  to  pursue  mathematics.  He  worked  quite 
independently  of  his  teachers,  and  while  a  student  at  Gottingen  made 
several  of  his  greatest  discoveries.  Higher  arithmetic  was  his  favorite 
study.  Among  his  small  circle  of  intimate  friends  was  Wolfgang 
Bolyai.  After  completing  his  course  he  returned  to  Brunswick.  In 
1798  and  1799  he  repaired  to  the  university  at  Helmstadt  to  consult 
the  librarys  and  there  made  the  acquaintance  of  J.  F.  Pfaff,  a  mathe- 
matician of  much  power.  In  1S07  the  Emperor  of  Russia  ofTered  Gauss 
a  chair  in  the  Academy  at  St.  Petersburg,  but  by  the  advice  of  the 
astonomer  Olbers,  who  desired  to  secure  him  as  director  of  a  proposed 
new  observatory  at  Gottingen,  he  decHned  the  offer,  and  accepted 
the  place  at  Gottingen.  Gauss  had  a  marked  objection  to  a  mathe- 
matical chair,  and  preferred  the  post  of  astronomer,  that  he  might 
give  all  his  time  to  science.  He  spent  his  life  in  Gottingen  in  the  midst 
of  continuous  work.  In  1828  he  went  to  Berlin  to  attend  a  meeting 
of  scientists,  but  after  this  he  never  again  left  Gottingen,  except  in 
1854,  when  a  railroad  was  opened  between  Gottingen  and  Hanover. 
He  had  a  strong  will,  and  his  character  showed  a  curious  mixture  of 
self-conscious  dignity  and  child-like  simplicity.  He  was  little  com- 
municative, and  at  times  morose.  Of  Gauss'  collected  works,  or 
Werke,  an  eleventh  volume  was  planned  in  19 16,  to  be  biographical  and 
bibliographical  in  character. 

A  new  epoch  in  the  theory  of  numbers  dates  from  the  publication 
of  his  Disquisitiones  ArithmeticcB,  Leipzig,  iSoi.  The  beginning  of 
this  work  dates  back  as  far  as  1795.  Some  of  its  results  had  been 
previously  given  by  J.  Lagrange  and  L.  Euler,  but  were  reached  inde- 
pendently by  Gauss,  who  had  gone  deeply  into  the  subject  before  he 
became  acquainted  with  the  writings  of  his  great  predecessors.  The 
Disquisitiones  ArithmeticcB  was  already  in  print  when  A.  M.  Legendre's 
Theorie  des  N ombres  appeared.  The  great  law  of  quadratic  reciprocity, 
given  in  the  fourth  section  of  Gauss'  work,  a  law  which  involves  the 
whole  theory  of  quadratic  residues,  was  discovered  by  him  by  in- 
duction before  he  was  eighteen,  and  was  proved  by  him  one  year 
later.  Afterwards  he  learned  that  L.  Euler  had  imperfectly  enunciated 
that  theorem,  and  that  A.  M.  Legendre  had  attempted  to  prove  it, 

^  Gauss^  wissensckafllicke  Tagebuck,  1796-18 14.  ^lit  Anmerkungen  herausgege- 
ien  von  Felix  Klein,  Berlin,  1901. 


436  A  HISTORY  OF  AIATHEMATICS 

but  met  with  apparently  insuperable  difficulties.  In  the  fifth  section 
Gauss  gave  a  second  proof  of  this  "gem"  of  higher  aritlmietic.  In 
1808  followed  a  third  and  fourth  demonstration;  in  181 7,  a  fifth  and 
sixth.  No  wonder  that  he  felt  a  personal  attachment  to  this  theorem. 
Proofs  ^  were  given  also  by  C.  G.  J.  Jacobi,  F.  Eisenstein,  J.  Liouville, 
Victor  Amedee  Lebesgue  (1791-1S75)  of  Bordeaux,  Angelo  Genocchi 
(1817-1889)  of  the  University  of  Turin,  E.  E.  Kummer,  M.  A.  Stern, 
Christian  Zeller  (1822-1899)  of  Markgroningen,  L.  Kronecker,  Victor 
Jacovlevich  Bouniakovski  (1S04-1SS9)  of  Petrograd,  Ernst  Schering 
(1833-1S97)  of  Gottingen,  Julius  Peter  Christian  Petersen  (1839-1910) 
of  Copenhagen,  E.  Busche,  Th.  Pepin,  Fabian  Franklin,  J.  C.  Fields, 
and  others.  Quadratic  reciprocity  "stands  out  not  only  for  the  in- 
fluence it  has  exerted  in  many  branches,  but  also  for  the  number  of 
new  methods  to  which  it  has  given  birth"  (P.  A.  MacMahon).  The 
solution  of  the  problem  of  the  representation  of  numbers  by  binary 
quadratic  forms  is  one  of  the  great  achievements  of  Gauss.  He  created 
a  new  algorithm  by  introducing  the  theory  of  congruences.  The  fourth 
section  of  the  Disquisiiiones  Arithmetica;,  treating  of  congruences  of 
the  second  degree,  and  the  fifth  section,  treating  of  quadratic  forms, 
were,  until  the  time  of  C.  G.  J.  Jacobi,  passed  over  with  universal 
neglect,  but  they  have  since  been  the  starting-point  of  a  long  series 
of  important  researches.  The  seventh  or  last  section,  developing  the 
theory  of  the  division  of  the  circle,  was  received  from  the  start  with 
deserved  enthusiasm,  and  has  since  been  repeatedly  elaborated  for 
students.  A  standard  work  on  Kreistheilimg  was  published  in  1872 
by  Paul  Bachmann,  then  of  Breslau. 

The  equation  for  the  division  of  the  circle  and  the  construction  of 
a  regular  polygon  of  n  sides,  n  being  prime,  can  be  solved  by  square 
root  extractions  alone,  always  and  only  when  n  —  i  is  a  power  of  2. 
Hence  such  regular  polygons  can  be  constructed  by  ruler  and  com- 
passes when  the  prime  number  n  is  3,  5,  17,  257,  65,537,  •  •  but  cannot 
be  constructed  when  w  is  7,  11,  13,  .  .  The  results  may  be  stated  also 
thus:  The  Greeks  knew  how  to  inscribe  regular  polygons  whose  sides 
numbered  2"*,  2"'.  3,  2'".  5  and  2™.  15.  Gauss  added  in  iSoi  that  the 
construction  is  possible  when  the  number  of  sides  n  is  prime  and  of 
the  form  2^'*+i.  L.  E.  Dickson  computed  that  the  number  of  such 
inscriptible  polygons  for  n  <  100  is  24,  for  n  <  300  is  37,  for  n  <  1000  is 
52,  for  «  <  100,000  is  206. 

Three  classical  constructions  of  the  regular  inscribed  polygon  of 
seventeen  sides  have  been  given:  one  by  J.  Serret  in  his  Algebra,  II, 
§  547,  another  by  von  Staudt  in  Crelle,  Vol.  24,  and  a  third  by  L. 
Gerard  in  Math.  Ainialcii,^  Vol.  4S  (1897),  using  compasses  only.  The 
analytic  solution,  as  outlined  by  Gauss,  was  actually  carried  out  for 
the  regular  polygon  of  257  sides  by  F.  J.  Richelot  of  Konigsberg  in 
four  articles  in  Crclle,  Vol.  9.  For  the  ])olygon  of  65,537  sides  this 
*  O.  Baumgart,  Ucbcr  das  Quadralischt:  Rcci procitatsgcsdz,  Lei])zig,  1885. 


THEORY  OF  NUMBERS  43^ 

was  accomplished  after  ten  years  of  labor  by  Oswald  Hermes  (1826- 
1909)  of  Steglitz;  his  manuscript  is  deposited  in  the  mathematical 
seminar  at  Gottingen.^  Gauss  had  planned  an  eighth  section  of  his  Dis 
quisitioncs  Arithmeticae,  which  was  omitted  to  lessen  the  expense  of 
jHiblication.  His  papers  on  the  theory  of  numbers  were  not  all  included 
in  his  great  treatise.  Some  of  them  were  published  for  the  first  time 
after  his  death  in  his  collected  works.  He  wrote  two  memoirs  on  the 
theor}^  of  biquadratic  residues  (1S25  and  1831),  the  second  of  which 
contains  a  theorem  of  biquadratic  reciprocity. 

K.  F.  Gauss  was  led  to  astronomy  by  the  discovery  of  the  planet 
Ceres  at  Palermo  in  1801.  His  determination  of  the  elements  of  its 
orbit  with  sufficient  accuracy  to  enable  H.  W.  M.  Olbers  to  rediscover 
it,  made  the  name  of  Gauss  generally  known.  In  1809  he  published 
the  Thcoria  mollis  corponim  ccelcslium,  which  contains  a  discussion 
of  the  problems  arising  in  the  determination  of  the  movements  of 
planets  and  comets  from  observations  made  on  them  under  any  cir- 
cumstances. In  it  are  found  four  formula;  in  sj^herical  trigonometry, 
now  usually  called  "Gauss'  Analogies,"  but  which  were  published 
somewhat  earlier  by  Karl  Brandon  Mollweide  (1774-1S25)  of  Leipzig, 
and  earlier  still  by  Jean  Baptiste  Joseph  Delambre  (1749-1822).^ 
IVlany  years  of  hard  work  were  spent  in  the  astronomical  and  magnetic 
obser\'ator}'.  He  founded  the  German  Magnetic  Union,  with  the 
object  of  securing  continuous  observations  at  fixed  times.  He  took 
part  in  geodetic  obser\-ations,  and  in  1843  and  1S46  wrote  two  me- 
moirs, Ueber  Gcgenstdnde  dcr  hohcren  Geodesie.  He  wrote  on  the  at- 
traction of  homogeneous  ellipsoids,  1813.  In  a  memoir  on  capillary 
attraction,  1833,  he  solves  a  problem  in  the  calculus  of  variations 
involving  the  variation  of  a  certain  double  integral,  the  limits  of  in- 
t.gration  being  also  variable;  it  is  the  earliest  example  of  the  solution 
of  such  a  problem.  He  discussed  the  problem  of  rays  of  light  passing 
through  a  system  of  lenses. 

Among  Gauss'  pupils  were  Heinrich  Christian  Schumacher,  Chris- 
tian Gerling,  Friedrich  Nicolai,  August  Ferdinand  Mobius,  Georg 
Wilhelm  Struve,  Johann  Frantz  Encke. 

Gauss'  researches  on  the  theor}^  of  numbers  were  the  starting-point 
for  a  school  of  writers,  among  the  earliest  of  whom  was  C.  G.  J. 
Jicobi.  The  latter  contributed  to  Crdle's  Journal  an  article  on  cubic 
residues,  giving  theorems  without  proofs.  After  the  publication  of 
Gauss'  pai)er  on  biquadratic  residues,  giving  the  law  of  biquadratic 
reciprocity,  and  his  treatment  of  complex  numbers,  C.  G.  J.  Jacobi 
found  a  similar  law  for  cubic  residues.  By  the  theory  of  elliptical 
functions,  he  was  led  to  beautiful  theorems  on  the  representation  of 

'  A.  Mitzscherling,  Das  Problem  dcr  KrcistcUung,  Leipzig  u  Berlin,  1913.  pp.  14., 

23- 

-I.  Todhunter,  "Note  on  the  History  of  Certain  Formulae  in  Sj)herical  Trigo- 
nometry," Philosophical  Magazine,  Feb.,  1S73. 


438  A  HISTORY  OF  MATHEMATICS 

numbers  by  2,  4,  6,  and  8  squares.  Next  come  the  researches  of  P.  G. 
L.  Dirichlet,  the  expounder  of  Gauss,  and  a  contributor  of  rich  results 
of  his  own. 

Peter  Gustav  Lejeune  Dirichlet  *  (1805-1859)  was  bom  in  Diiren, 
attended  the  gymnasium  in  Bonn,  and  then  the  Jesuit  gymnasium 
in  Cologne.  In  1822  he  was  attracted  to  Paris  by  the  names  of  P.  S. 
Laplace,  A.  M.  Legendre,  J.  Fourier,  D.  S.  Poisson,  and  A.  L.  Cauchy. 
The  facilities  for  a  mathematical  education  there  were  far  better 
than  in  Germany,  where  K.  F.  Gauss  was  the  only  great  figure.  He 
read  in  Paris  Gauss'  Disquisitiones  Arithmeticce,  a  work  which  he 
never  ceased  to  admire  and  study.  Much  in  it  was  simplified  by 
Dirichlet,  and  thereby  placed  within  easier  reach  of  mathematicians. 
His  first  memoir  on  the  impossibility  of  certain  indeterminate  equa- 
tions of  the  fifth  degree  was  presented  to  the  French  Academy  in  1825. 
He  showed  that  P.  Fermat's  equation,  x^+y'^=z'^,  cannot  exist  when 
w=5.  Some  parts  of  the  analysis  are,  however,  A.  M.  Legendre's. 
Dirichlet's  acquaintance  with  J.  Fourier  led  him  to  investigate  Four- 
ier's series.  He  became  docent  in  Breslau  in  1827.  In  1828  he  ac- 
cepted a  position  in  Berlin,  and  finally  succeeded  K.  F.  Gauss  at 
Gottingen  in  1855.  The  general  principles  on  which  depends  the 
average  number  of  classes  of  binary  quadratic  forms  of  positive  and 
negative  determinant  (a  subject  first  investigated  by  Gauss)  were 
given  by  Dirichlet  in  a  memoir,  Ueber  die  Bestimmung  der  mittleren 
Werthe  in  der  Zahlentheorie,  1849.  More  recently  F.  Mertens  of  Graz, 
since  1894  of  Vienna,  determined  the  asymptotic  values  of  several 
numerical  functions.  Dirichlet  gave  some  attention  to  prime  num- 
bers. K.  F.  Gauss  and  A.  M.  Legendre  had  given  expressions  denoting 
approximately  the  asymptotic  value  of  the  number  of  primes  inferior 
to  a  given  limit,  but  it  remained  for  G.  F.  B.  Riemann  in  his  memoir, 
Ueber  die  Anzahl  der  Primzahlen  unter  einer  gegebenen  Grosse,  1859, 
to  give  an  investigation  of  the  asymptotic  frequency  of  primes  which 
is  rigorous.  Approaching  the  problem  from  a  different  direction, 
P.  L.  Chebichev,  formerly  professor  in  the  University  of  St.  Petersburg, 
established,  in  a  celebrated  memoir,  Sur  les  N ombres  Premiers,  1850, 
the  existence  of  limits  within  which  the  sum  of  the  logarithms  of  the 
primes  P,  inferior  to  a  given  number  x,  must  be  comprised.^  He 
proved  that,  if  ?z>3,  there  is  always  at  least  one  prime  between  n 
and  2n—2  (inclusive).  This  theorem  is  sometimes  called  "Bertrand's 
postulate,"  since  J.  L.  F.  Bertrand  had  previously  assumed  it  for 
the  purpose  of  proving  a  theorem  in  the  theory  of  substitution  groups. 
This  paper  depends  on  very  elementary  considerations,  and,  in  that 
respect,  contrasts  strongly  with  Riemann's,  which  involves  abstruse 
theorems  of  the  integral  calculus.     H.  Poincare's  papers,  J.  J.  Syl- 

^  E.  E.  Kummer,  Gedachtnissrede  anf  Gustav  Peter  Lejeiine-Dirichlet,  Berlin,  i860. 
2  H.  J.  Stephen  Smith  "On  the  Present  State  and  Prospects  of  some  Branches  of 
Pure  Mathematics,"  Proceed.  London  Math.  Soc,  Vol.  8,  1876,  p.  17. 


THEORY  OF  NUMBERS  439 

vester's  contraction  of  Chebichev's  limits,  with  reference  to  the  dis- 
tribution of  primes,  and  researches  of  J.  Hadamard  (awarded  the 
Grand  prix  of  1S92),  are  among  the  later  researches  in  this  line. 
G.  F.  B.  Riemann  had  advanced  six  properties  relating  to 
00     I 
^  (5)  =  S    — ,  where  s=  a+ti, 

none  of  which  he  was  able  to  prove. ^  In  1893  J.  Hadamard  proved 
three  of  these,  thereby  establishing  the  existence  of  null-places  in 
Riemann's  zeta-f unction;  H.  von  Mangoldt  of  Danzig  proved  in  1895 
a  fourth  and  in  1905  a  tifth  of  Riemann's  six  properties.  The  remain- 
ing one,  that  the  roots  of  C(s)  in  the  strip  oS  J  <  1,  have  all  the  real 
part  I,  remains  unproved,  though  progress  in  the  study  of  this  case 
has  been  made  by  F.  Mertens  and  R.  v.  Sterneck.  If  x  is  a  positive 
number,  and  if  7r(.r)  denotes  the  number  of  primes  less  than  .r,  then 
what  Landau  calls  the  "prime-number  theorem"  (Primzahlsatz) 
states  that  the  ratio  of  7r(.v)  to  x/\og  x  approaches  i  as  x  increases 
without  end.  A.  M.  Legendre,  K.  F.  Gauss,  and  P.  G.  L.  Dirichlet 
had  guessed  this  theorem.  As  early  as  1737  L.  Euler'-  had  given  an 
analogous  theorem,  that  Zi'p  approaches  log  (log  p),  where  the  sum- 
mation extends  over  all  primes  not  greater  than  p.  The  prime-number 
theorem  was  proved  in  1896  by  J.  Hadamard  and  Charles  Jean  de  la 
Vallee  Poussin  of  Louvain,  in  1901  by  Nils  Fabian  Helge  von  Koch 
of  Stockholm,  in  1903  by  E.  Landau,  now  of  Gottingen,  in  1915  by 
G.  H.  Hardy  and  J.  E.  Littlewood  of  Cambridge.  Hardy  discovered 
an  infinity  of  zeroes  of  the  zeta-f  unction  with  the  real  part  ^;  E. 
Landau  simplified  Hardy's  proof. 

G.  F.  B.  Riemann's  zeta  function  t,{s)  was  first  studied  on  account 
of  its  fundamental  importance  in  the  theory  of  prime  numbers,  but  it 
has  become  important  also  in  the  theory  of  analytic  functions  in 
general.  In  1909  E.  Landau  published  his  Handbuch  der  Lehre  von 
der  Verteilung  der  Primzahlen.  In  191 2  he  pronounced  the  following 
four  questions  to  be  apparently  incapable  of  answer  in  the  present 
state  of  the  science  of  numbers:  (i)  Does  ;r+i  for  integral  values  of 
n  represent  an  infinite  number  of  primes?  (2)  C.  Goldbach's  theorem: 
Can  prime  values  of  p  and  p'  be  found  to  satisfy  m  =  p+p'  for  each 
even  m  larger  than  2?  (3)  Has  2=p  —  p^  an  infinite  number  of  solutions 
in  primes?  (4)  Is  there  between  n^  and  (n+i)-  at  least  one  prime  for 
every  positive  integral  n? 

The  enumeration  of  prime  numbers  has  been  undertaken  at  differ- 
ent times  by  various  mathematicians.  Factor  tables,  giving  the  least 
factor  of  every  integer  not  divisible  by  2,  3,  or  5,  did  not  extend  above 
408,000  previous  to  the  year  181 1,  when  Ladislaus  Chernac  published 
his  Cribrum  arilhmeticiim  at  Deventer  in  Netherlands,  which  gives 

'  For  details,  consult  E.  Landau  in  Proceed,  jth  Intern.  Congress,  Cambridge,  19 12, 
Vol.  I,  1913,  p.  97. 

2  G.  Enestrom  in  Bibliotheca  mathemalka,  3.  S.,  Vol.  i,^,  p.  81. 


440  A  HISTORY  OF  MATHEMATICS 

factors  for  numbers  up  to  1,020,000.  J.  Ch.  Burckhardt  (1773-1815) 
published  factor  tables  in  Paris,  in  1S17  for  the  numbers  i  to  1,020,000, 
in  1814  for  the  numbers  1020000  to  2028000.  in  1816  for  the  numbers 
2,028,000  to  3,036,000.  James  Glaisher  (1809-1903)  pubhshed  factor 
tables  at  London,  in  1879  for  the  numbers  3,000,000  to  4,000,000,  in 
1880  for  numbers  4,000,000  to  5,000,000,  in  1883  for  the  numbers 
5,000,000  to  6,000,000.  Zacharias  Dase  (1824-1861)  published  factor 
tables  at  Hamburg,  in  1862  for  the  numbers  6,000,001  to  7,002,000,  in 
1863  for  the  numbers  7,002,001  to  8,010,000,  in  1865  for  the  numbers 
8  oio.ooi  to  9,000,000.  In  1909  the  Carnegie  Institution  of  Washing- 
ton published  factor  tables  for  the  first  ten  millions,  prepared  by  D. 
N.  Lehmer  of  the  University  of  California.  Lehmer  gives  the  errors 
discovered  in  the  earlier  publications.  Historical  details  about  factor 
tables  are  given  by  Glaisher  in  his  Factor  Table.  Fourth  Million,  1879. 
Miscellaneous  contributions  to  the  theory  of  numbers  were  made 
by  ^.  L.  Caiichy.  He  showed,  for  instance,  how  to  find  all  the  infinite 
solutions  of  a  homogeneous  indeterminate  equation  of  the  second 
degree  in  three  variables  when  one  solution  is  given.  He  established 
the  theorem  that  if  two  congruences,  which  have  the  same  modulus, 
admit  of  a  common  solution,  the  modulus  is  a  divisor  of  their  resultant. 
Joseph  Liouviile  (1809- 1882),  professor  at  the  College  de  France, 
investigated  mainly  questions  on  the  theory  of  quadratic  forms  of  two, 
and  of  a  greater  number  of  variables.  A  research  along  a  different 
line  proved  to  be  an  entering  wedge  into  a  subject  which  since  has 
become  of  vital  importance.  In  1844  he  proved  {Liouviile' s  Journal, 
Vol.  5)  that  neither  e  nor  e^  can  be  a  root  of  a  quadratic  equation  with 
rational  coefficients.  By  the  properties  of  convergents  of  a  continued 
fraction  representing  a  root  of  an  algebraical  equation  with  rational 
coefficients  he  established  later  the  existence  of  numbers — the  so- 
called  transcendental  numbers — -which  cannot  be  roots  of  any  such 
equation.  He  proved  this  also  by  another  method.  A  still  different 
approach  is  due  to  G.  Cantor.  Profound  researches  were  instituted 
by  Ferdinand  Gotthold  Eisenstein  (1823-1852),  of  Berlin.  Ternary 
quadratic  forms  had  been  studied  somewhat  by  K.  F.  Gauss,  but  the 
extension  from  two  to  three  indeterminates  was  the  work  of  Eisen- 
stein who,  in  his  memoir,  Neue  Theoreme  der  hoheren  Arithmetic, 
defined  the  ordinal  and  generic  characters  of  ternary  quadratic  forms 
of  uneven  determinant;  and,  in  case  of  definite  forms,  assigned  the 
weight  of  any  order  or  genus.  But  he  did  not  publish  demonstrations 
of  his  results.  In  inspecting  the  theory  of  binary  cubic  forms,  he  was 
led  to  the  discovery  of  the  first  covariant  ever  considered  in  analysis. 
He  showed  that  the  series  of  theorems,  relating  to  the  presentation 
of  numbers  by  sums  of  squares,  ceases  when  the  number  of  squares 
surpasses  eight.  Many  of  the  proofs  omitted  by  Eisenstein  were  sup- 
plied by  Henry  Smith,  who  was  one  of  the  few  Englishmen  who  de- 
^'oted  themselves  to  the  study  of  higher  arithmetic. 


THEORY  OF  NUMBERS  441 

Henry  John  Stephen  Smith  ^  (1826-18S3)  was  born  in  London, 
and  educated  at  RuGjby  and  at  Balliol  College,  Oxford.  Before  1847 
he  travelled  much  in  Europe  for  his  health,  and  at  one  time  attended 
lectures  of  D.  F.  J.  Arago  in  Paris,  but  after  that  year  he  was  never 
absent  from  Oxford  for  a  single  term.  In  1849  he  carried  ofT  at  Oxford 
the  highest  honors,  both  in  the  classics  and  in  mathematics,  thus 
ranking  as  a  "double  first."  There  is  a  story  that  he  decided  between 
classics  and  mathematics  as  the  field  for  his  life-work,  by  tossing  up  a 
penny.  He  never  married  and  had  no  household  cares  to  destroy  the 
needed  serenity  for  scientific  work,  "excepting  that  he  was  careless  in 
money  matters,  and  trusted  more  to  speculation  in  mining  shares 
than  to  economic  management  of  his  income."  -  In  1861  he  was 
elected  Savilian  professor  of  geometry.  His  first  paper  on  the  theory 
of  numbers  appeared  in  1S55.  The  results  of  ten  years'  study  of 
everything  published  on  the  theory  of  numbers  are  contained  in  his 
Reports  which  appeared  in  the  British  Association  volumes  from  1859 
to  1865.  These  reports  are  a  model  of  clear  and  precise  exposition 
and  perfection  of  form.  They  contain  much  original  matter,  but  the 
chief  results  of  his  own  discoveries  were  printed  in  the  Philosophical 
Transactions  for  1861  and  1867.  They  treat  of  Hnear  indeterminate 
equations  and  congruences,  and  of  the  orders  and  genera  of  ternary 
quadratic  forms.  He  established  the  principles  on  which  the  exten- 
sion to  the  general  case  of  n  indeterminates  of  quadratic  forms  de- 
pends. He  contributed  also  two  memoirs  to  the  Proceedings  of  the 
Royal  Society  of  1864  and  1868,  in  the  second  of  which  he  remarks  that 
the  theorems  of  C.  G.  J.  Jacobi,  F.  Eisenstein,  and  J.  Liouville,  re- 
lating to  the  representation  of  numbers  by  4,  6,  8  squares,  and  other 
simple  quadratic  forms  are  deducible  by  a  uniform  method  from  the 
principles  indicated  in  his  paper.  Theorems  relating  to  the  case  of 
5  squares  were  given  by  F.  Eisenstein,  but  Smith  completed  the  enunci- 
ation of  them,  and  added  the  corresponding  theorems  for  7  squares. 
The  solution  of  the  cases  of  2,  4,  6  squares  may  be  obtained  by  elliptic 
functions,  but  when  the  number  of  squares  is  odd,  it  involves  processes 
peculiar  to  the  theory  of  numbers.  This  class  of  theorems  is  limited 
to  8  squares,  and  Smith  completed  the  group.  In  ignorance  of  Smith's 
investigations,  the  French  Academy  ofTered  a  prize  for  the  demon- 
stration and  completion  of  F.  Eisenstein's  theorems  for  5  squares. 
This  Smith  had  accomplished  fifteen  years  earlier.  He  sent  in  a  dis- 
sertation in  1882,  and  next  year,  a  month  after  his  death,  the  prize 
was  awarded  to  him,  another  prize  being  also  awarded  to  H.  Min- 
kowsky of  Bonn.  The  theory  of  numbers  led  Smith  to  the  study  of 
elliptic  functions.  He  wrote  also  on  modern  geometry.  His  succes- 
sor at  Oxford  was  J.  J.  Sylvester.    Taking  an  anti-utilitarian  view  ot 

>  J.  W.  L.  Glaisher  in  Monthly  Notices  R.  Astr.  Soc,  Vol.  44,  1884. 
2  A.  Macfarlane,  Ten  British  Mathematicians,  igiO,  p.  98. 


442  A  HISTORY  OF  MATHEMATICS 

mathematics,    Smith   once   proposed   a   toast,    "Pure  mathematics; 
may  it  never  be  of  any  use  to  any  one." 

Ernst  Eduard  Kummer  (1S10-1893),  professor  in  the  University 
of  Berlin,  is  closely  identified  with  the  theory  of  numbers.  P.  G.  L. 
Dirichlet's  work  on  complex  numbers  of  the  form  a+ib,  introduced 
by  K.  F.  Gauss,  was  extended  by  him,  by  F.  Eisenstein,  and  R.  Dede- 
kind.  Instead  of  the  equation  x^— 1=0,  the  roots  of  which  yield 
Gauss'  units,  F.  Eisenstein  used  the  equation  ;v^— 1=^0  and  complex 
numbers  a+bp  (p  being  a  cube  root  of  unity),  the  theory  of  which 
resembles  that  of  Gauss'  numbers.  E.  E.  Kummer  passed  to  the 
general  case  x"  —  1=0  and  got  com.plex  numbers  of  the  form  a  =  aiA  1+ 
a-zAo+o-zA  3+  . . . ,  were  a,  are  v/hole  real  numbers,  and  A;  roots  of  the 
above  equation.  Euclid's  theory  of  the  greatest  common  divisor  is 
not  applicable  to  such  complex  numbers,  and  their  prime  factors  can- 
not be  defined  in  the  same  way  as  prime  factors  of  common  integers 
are  defined.  In  the  effort  to  overcome  this  difhculty,  E.  E.  Kummer 
was  led  to  introduce  the  conception  of  "ideal  numbers."  These 
ideal  numbers  have  been  applied  by  G.  Zolotarev  of  St.  Petersburg 
to  the  solution  of  a  problem  of  the  integral  calculus,  left  unfinished  by 
Abel.^  J.  W.  R.  Dedekind  of  Braunschweig  has  given  in  the  second 
edition  of  Dirichlet's  Vorlcsungen  iiber  Zahlentheorie  a  new  theory 
of  complex  numbers,  in  which  he  to  some  extent  deviates  from  the 
course  of  E.  E.  Kummer,  and  avoids  the  use  of  ideal  numbers.  De- 
dekind has  taken  the  roots  of  any  irreducible  equation  with  integral 
coefficients  as  the  units  for  his  complex  numbers.  F.  Klein  in  1893 
introduced  simplicity  by  a  geometric  treatment  of  ideal  numbers. 

Fermat's  "Last  Theorem,"  Waring's  Theorem 

E.  E.  Kummer's  ideal  numbers  owe  their  origin  to  his  efforts  to 
prove  the  impossibility  of  solving  in  integers  Fermat's  equation 
xn-\-yn=2^  for  n>2.  We  premise  that  some  progress  in  proving  this 
impossibility  has  been  made  by  more  elementary  means.  For  in- 
tegers X,  y,  z  not  divisible  by  an  odd  prime  n,  the  theorem  has  been 
proved  by  the  Parisian  mathematician  and  philosopher  Sophie  Ger- 
main (1776-1831)  for  ;r<ioo,  by  Legendre  lor  «<2oo,  by  E.  T.  Mail- 
let  for  w<223,  by  Dmitry  Mirimanoff  for  n<2$-],  by  L.  E.  Dickson 
for  nK'jooo?  The  method  used  here  is  due  to  Sophie  Germain  and 
requires  the  determination  of  an  odd  prime  p  for  which  x«+y"+ 
j:"^o  (mod.  p)  has  no  solutions,  each  not  divisible  by  p,  and  n  is  not 
the  residue  modulo  p  of  the  nth.  power  of  any  integer.  E.  E.  Kum- 
mer's results  rest  on  an  advanced  theory  of  algebraic  numbers  which  he 

1  H.  J.  S.  Smith,  "On  the  Present  State  and  Prospects  of  Some  Branches  of  Pure 
Mathematics,"  Proceed.  London  Maih.  Soc,  Vol.  8,  1876,  p.  15. 

2  See  L.  E.  Dickson  in  Annals  of  Mathemalics,  2.  S.,  Vol.  18,  1917,  pp.  161-187. 
See  also  L.  E.  Dickson  in  Alii  del  IV.  Congr.  Roma,  1008,  Roma,  1909,  Vol.  11, 
p.  172. 


THEORY  OF  NUMBERS  "  44^^ 

helped  to  create.  Once  at  an  early  period  he  thought  that  he  had  a  com- 
plete proof.  He  laid  it  before  P.  G.  L.  Dirichlet  who  pointed  out  that, 
although  he  had  proved  that  any  number /(a),  where  a  is  a  complex  n"' 
root  of  unity  and  ;/  is  prime,  was  the  product  of  indecomposable  factors, 
he  had  assumed  that  such  a  factorization  was  unique,  whereas  this  was 
not  true  in  general.^  After  years  of  study,  E.  E.  Kummer  concluded 
that  this  non-uniqueness  of  factorization  was  due  to /(a)  being  too 
small  a  domain  of  numbers  to  permit  the  presence  in  it  of  the  true  prime 
numbers.  He  was  led  to  the  creation  of  his  ideal  numbers,  the  ma- 
chinery of  which,  says  L.  E.  Dickson,^  is  "so  delicate  that  an  expert 
must  handle  it  with  the  greatest  care,  and  (is)  nowadays  chiefly  of 
historical  interest  in  view  of  the  simpler  and  more  general  theory  of 
R.  Dedekind."  By  means  of  his  ideal  numbers  he  produced  a  proof 
of  Fermat's  last  theorem,  which  is  not  general  but  excludes  certain 
particular  values  of  n,  which  values  are  rare  among  the  smaller  values 
of  n;  there  are  no  values  of  n  below  loo,  for  which  E.  E.  Kummer's 
proof  does  not  serve.  In  1857  the  French  Academy  of  Sciences 
awarded  E.  E.  Kummer  a  prize  of  3000  francs  for  his  researches  on 
complex  integers. 

The  first  marked  advance  since  Kummer  was  made  by  A.  Wieferich 
of  Miinster,  in  Crelle's  Journal,  Vol.  136,  1909,  who  demonstrated  that 
if  p  is  prime  and  2^  —  2  is  not  divisible  by  p"^,  the  equation  xP+yP=zP 
cannot  be  solved  in  terms  of  positive  integers  which  are  not  mul- 
tiples of  p.  Waldemar  Meissner  of  Charlottenburg  found  that  2^  —  2 
is  divisible  by  p"^  when  p  =  iog7,  and  for  no  other  prime  p  less  than 
2000.  Recent  advances  toward  a  more  general  proof  of  Fermat's 
last  theorem  have  been  made  by  D.  Mirimanoff  of  Geneva,  G.  Fro- 
benius  of  Berlin,  E.  Hecke  of  Gottingen,  F.  Bernstein  of  Gottingen, 
Ph.  Furtwiingler  of  Bonn,  S.  Bohnicek  and  H.  S.  Vandiver  of  Phila- 
delphia. Recent  efforts  along  this  line  have  been  stimulated  in  part 
by  a  bequest  of  100,000  marks  made  in  1908  to  the  Konigliche  Gesell- 
schaft  der  Wissenschaften  in  Gottingen,  by  the  mathematician  F.  P. 
Wolfskehl  of  Darmstadt,  as  a  prize  for  a  complete  proof  of  Fermat's 
last  theorem.  Since  then  hundreds  of  erroneous  proofs  have  been 
published.  Post-mortems  over  proofs  which  fall  still-born  from  the 
press  are  being  held  in  the  "  Sprechsaal"  of  the  Archiv  der  Mathematik 
und  Physik. 

At  the  beginning  of  the  present  century  progress  was  made  in  prov- 
ing another  celebrated  theorem,  known  as  "Waring's  theorem."  In 
1909  A.  Wieferich  of  Miinster  proved  the  part  which  says  that  every 
positive  integer  is  equal  to  the  sum  of  not  more  than  9  positive  cubes. 
He  established  also,  that  every  positive  integer  is  equal  to  the  sum 
of  not  more  than  37  (according  to  Waring,  it  is  not  more  than  19) 
positive  fourth  powers,  while  D.  Hilbert  proved  in  1909  that,  foi 

'  Festschrift  z.    Feierdes  loo.    Gcburtstages  Ediiard  Kummcrs,  Leipzig,  19 10,  p.  2? 
-Bull.  Am.  Math.  Soc,  Vol.  17,  1911,  p.  371. 


444  A  HISTORY  OF  MATHEMATICS 

every  integer  «>2  (Waring  had  declared  for  every  integer  n>4).^ 
each  positive  integer  is  expressible  as  the  sum  of  positive  nth.  powers, 
the  number  of  which  lies  within  a  limit  dependent  only  upon  the 
value  of  II.  Actual  determinations  of  such  upper  limits  have  been 
made  by  A.  Hurwitz,  E.  T.  Maillet,  A.  Fleck,  and  A.  J.  Kempner, 
Kempner  proved  in  191 2  that  there  is  an  infinity  of  numbers  which 
are  not  the  sum  of  less  than  4  .  2"  positive  2«th  powers,  n^2. 

Other  Recent  Researches.    Number  Fields 

Attracted  by  E.  E.  Kummer's  investigations,  his  pupil,  Leopold 
Kronecker  (1823-1891)  made  researches  which  he  applied  to  algebraic 
equations.  On  the  other  hand,  efforts  have  been  made  to  utilize  in 
the  theory  of  numbers  the  results  of  the  modern  higher  algebra. 
Following  up  researches  of  Ch.  Hermite,  Paul  Bachmann  of  Mimster, 
now  of  Weimar,  investigated  the  arithmetical  formula  which  gives 
the  automorphics  of  a  ternary  quadratic  form.'  Bachmann  is  the 
author  of  well-known  texts  on  Zahlentheorie,  in  several  volumes,  which 
appeared  in  1892,  1894,  1872,  1898,  and  1905,  respectively.  The  prob- 
lem of  the  equivalence  of  two  positive  or  definite  ternary  quadratic 
forms  was  solved  by  L.  Seeber;  and  that  of  the  arithmetical  auto- 
morphics of  such  forms,  by  F.  G.  Eisenstein.  The  more  difficult  prob- 
lem of  the  equivalence  for  indefinite  ternary  forms  has  been  investi- 
gated by  Eduard  Selling  of  Wurzburg.  On  quadratic  forms  of  four 
or  more  indeterminates  little  has  yet  been  done.  Ch.  Hermite  showed 
that  the  number  of  non-equivalent  classes  of  quadratic  forms  having 
integral  coefficients  and  a  given  discriminant  is  finite,  while  Zolotarev 
and  Alexander  Korkine  (18 37-1 90S),  both  of  St.  Petersburg,  investi- 
gated the  minima  of  positive  quadratic  forms.  In  connection  with 
binary  quadratic  forms,  H.  J.  S.  Smith  established  the  theorem  that 
if  the  joint  invariant  of  two  properly  primitive  forms  vanishes,  the 
determinant  of  either  of  them  is  represented  primitively  by  the  dupli- 
cate of  the  other. 

The  interchange  -of  theorems  between  arithmetic  and  algebra  is 
displayed  in  the  recent  researches  of  J.  W.  L.  Glaisher  (1848-  ) 
of  Trinity  College  and  J.  J.  Sylvester.  Sylvester  gave  a  Constructive 
Theory  of  Partitions,  which  received  additions  from  his  pupils,  F. 
Franklin,  now  of  New  York  city,  and  George  Stetson  Ely  (?-i9i8), 
for  many  years  examiner  in  the  U.  S.  Patent  Office. 

By  the  introduction  of  "ideal  numbers"  E.  E.  Kummer  took  a 
first  step  toward  a  theory  of  fields  of  numbers.  The  consideration  of 
super  fields  (Oberkorper)  from  which  the  properties  of  a  given  field 
of  numbers  may  be  easily  derived  is  due  mainly  to  R.  Dedekind  and 
to  L.  Kronecker.  Thereby  there  was  opened  up  for  the  theory  of 
numbers  a  new  and  wide  territory  which  is  in  close  connection  with 
1  H.  J.  S.  Smith  in  Proceed.  London  Math.  Soc,  Vol.  8,  1876,  p.  13. 


THEORY  OF  NUMBERS  445 

algebra  and  the  theory  of  functions.  The  importance  of  this  subject 
in  the  theory  of  equations  is  at  once  evident  if  we  call  to  mind  E. 
Galois'  fields  of  rationality.  The  interrelation  between  number  theory 
and  function  theory  is  illustrated  in  Riemann's  researches  in  which 
the  frequency  of  primes  was  made  to  depend  upon  the  zero-places  of 
6.  certain  analytical  function,  and  in  the  transcendence  of  e  and  tt 
which  is  an  arithmetic  property  of  the  exponential  function.  In  18S3- 
1S90  L.  Kronecker  publisned  important  results  on  elliptic  functions 
which  contain  arithmetical  theorems  of  great  elegance.  The  Dedekind 
method  of  extending  Rummer's  results  to  algebraic  numbers  in 
general  is  based  on  the  notion  of  an  ideal.  A  common  characteristic 
of  Dedekind  and  Rronecker's  procedure  is  the  introduction  of  com- 
pound moduU.  G.  M.  Mathews  says  ^  that,  in  practice  it  is  convenient 
to  combine  the  methods  of  L.  Kronecker  and  R.  Dedekind.  Of 
central  importance  are  the  Galoisian  or  normal  fields,  which  have  been 
studied  extensively  by  D.  Hilbert.  L.  Kronecker  established  the 
theorem  that  all  Abelian  fields  are  cyclotomic,  which  was  proved  also 
by  H.  Weber  and  D.  Hilbert.  An  important  report,  prepared  by  D. 
Hilbert  and  entitled  Theorie  der  algebraischen  Zahlkirper,  was  pub- 
hshed  in  1894.-  D.  Hilbert  first  develops  the  theory  of  general  number- 
fields,  then  that  of  special  fields,  viz.,  the  Galois  field,  the  quadratic 
field,  the  circle  field  (Kreiskorper),  the  Kummer  field.  A  report  on 
later  investigations  was  published  by  R.  Fueter  in  igii.^  Chief  among 
the  workers  in  this  subject  which  have  not  yet  been  mentioned  are 
F.  Bernstein,  Ph.  Furtwiingler,  H.  Minkowski,  Ch.  Hermite,  and  A. 
Hurwitz.  Accounts  of  the  theory  are  given  in  H.  Weber's  Lehrbuch  der 
Algebra,  Vol.  2  (iSqq),  J.  Sommer's  Vorlesungen  uber  Zahlentheorie 
(1907),  and  Flermann  Minkowski's  Diophanlische  A  pproximationen, 
Leipzig  (1907).  H.  Minkowski  gives  in  geometric  and  arithmetic 
language  both  old  and  new  results.  His  use  of  lattices  serves  as  a 
geometric  setting  for  algebraic  theory  and  for  the  proof  of  some  new 
results. 

A  new  and  powerful  method  of  attacking  questions  on  the  theory 
of  algebraic  numbers  was  advanced  by  Kurt  Hensel  of  Konigsberg 
in  his  Theorie  der  algebraischen  Zahlen,  1908,  and  in  his  Zahlentheorie, 
1913.  His  method  is  analogous  to  that  of  power  series  in  the  theory 
of  analytic  functions.  He  employs  expansions  of  numbers  into  power 
series  in  an  arbitrary  prime  number  p.  This  theory  of  /?-adic  numbers 
is  generalized  by  him  in  his  book  of  1913  into  the  theory  of  g-adic 
numbers,  where  g  is  any  integer.^ 

The  resolution  of  a  given  large  number  into  factors  is  a  difficult 
problem  which  has  been  taken  up  by  Paul  Seelhof,  Frangois  Edouard 

^  .^rt.  "Xumlier"  in  the  Enryclof).  Brilainiica,  nth  ed.,  p.  857. 
^Jahresberichl  d.  d.  Math.  Vercinigimg,  Vol.  4,  pp.  177-546. 
^  Loc.  cit.,  Vol.  20,  )>p.  1-47. 
^BiUl.  Am.  Math.  Soc,  Vol.  20,  1914,  .j.  259. 


445  A  HISTORY  OF  MATHEMATICS 

Anatole  Lucas  (1842-1891)  of  Paris,  Fortune  Landry  (1799-?^  A.  J.  C 
Cunningham,  F.  W.  P.  Lawrence  and  D.  N.  Lehmer. 

Transcendental  Numbers.     The  Infinite 

Building  on  the  results  previously  reached  by  J.  Liouville,  Ch. 
Hermite  proved  in  1873  in  the  Comples  Rendus,  Vol.  77,  that  e  is 
transcendental,  while  F.  Lindemann  in  1882  {Ber.  Akad.  Berlin) 
proved  that  tt  is  transcendental.  Ch.  Hermite  reached  his  result  by 
showing  that  ae^-\-be'^-\-ce^+  ...  =0  cannot  subsist,  where  w,  n,  r, .  .  . 
a,  b,  c,  .  .  .  are  whole  numbers;  F.  Lindemann  proved  that  this  equa- 
tion cannot  subsist  when  in,  n,  r,  .  .  a,  b,  c  .  .  are  algebraic  numbers, 
that  in  particular,  e"+i=o  cannot  subsist  if  x  is  algebraic.  Conse- 
quently 7r  cannot  be  an  algebraic  number.  But,  starting  with  two 
points,  (o,  o)  and  (i,  o),  a  third  point  (a,  o)  can  be  constructed  by  tJie 
aid  of  ruler  and  compasses  only  when  a  is  a  certain  special  type  of 
algebraic  number  that  is  obtainable  by  successive  square  root  extrac- 
tions. Hence  the  point  (tt,  o)  cannot  be  constructed,  and  the  "quad- 
rature of  the  circle"  is  impossible.  The  proofs  of  Ch.  Hermite  and  F. 
Lindemann  involved  complex  integrations  and  were  complicated. 
Simplified  proofs  were  given  by  K.  Weierstrass  in  1885,  Th.  J.  Stieltjes 
in  1890,  D.  Hilbert,  A.  Hurwitz,  and  P.  Gordan  in  1S96  {Math.  An- 
nalen,  Vol.  43),  F.  Mertens  in  1896,  Th.  Vahlen  in  1900,  H.  Weber, 
F.  Enriques,  and  E.  W.  Hobson  in  191 1.  G.  B.  Halsted  says  of  the 
circle,  "John  Bolyai  squared  it  in  non-Euchdean  geometry  and  Linde- 
mann proved  no  man  could  square  it  in  Euclidean  geometry." 

That  there  are  many  other  transcendental  numbers  beside  e  and  ir 
is  evident  from  the  researches  of  J.  Liouville,  E.  Maillet,  G.  Faber 
and  Aubrey  J.  Kempner,  who  give  new  forms  of  infinite  series  which 
define  transcendental  numbers.  Of  interest  are  the  theorems  estab- 
lished in  1913  by  G.  N.  Bauer  and  H.  L.  Slobin  of  Minneapolis,  that 
the  trigonometric  functions  and  the  hyperbolic  functions  represent 
transcendental  numbers  whenever  the  argument  is  an  algebraic  num- 
ber other  than  zero,  and  vice  versa,  the  arguments  are  transcendental 
numbers  whenever  the  functions  are  algebraic  numbers.^ 

The  notions  of  the  actually  infinite  have  undergone  radical  change 
during  the  nineteenth  century.  As  late  as  183 1  K.  F.  Gauss  expressed 
himself  thus:  "I  protest  against  the  use  of  infinite  magnitude  as 
something  completed,  which  in  mathematics  is  never  permissible. 
Infinity  is  merely  a  fagon  de  parler,  the  real  meaning  being  a  limit 
which  certain  ratios  approach  indefinitely  near,  while  others  are  per- 
mitted to  increase  v/ithout  restriction."  ^  Gauss'  contemporary,  A.  L. 
Cauchy,  likewise  rejected  the  actually  infinite,  being  influenced  by 

'  Rendiconti  d.  Circolo  Math,  di  Palermo,  Vol.  38,  1914,  p.  353. 
2  C.  F.  Gauss,  Brief  on  Schumacher,  Werke,  Bd.  8,  216;  quoted  from  Moritz, 
Memorabilia  malhemaiica,  19 14,  p.  337. 


APPLIED  MATHEMAI'irs  447 

the  eighicentli  century  pliilosoplicr  of  Turin,  Father  GerdiL'  In  1SS6 
Georg  Cantor  occu[)ie(l  a  diametrically  opposite  position,  when  he 
said:  "In  spite  of  the  essential  dilTerence  between  the  conceptions  of 
the  potential  and  the  actual  infinite,  the  former  signifying  a  variable 
finite  magnitude  increasing  beyond  all  finite  limits,  while  the  latter 
is  a  fixed,  constant  quantity  lying  beyond  all  finite  magnitudes,  it 
hajipens  only  too  often  that  the  one  is  mistaken  for  the  other.  .  .  . 
Owing  to  a  justifiable  aversion  to  such  illegitimate  actual  infinities 
and  the  influence  of  the  modern  epicuric-materialistic  tendency,  a 
certain  horror  infiniti  has  grown  up  in  extended  scientific  circles, 
which  finds  its  classic  expression  and  support  in  the  letter  of  Gauss, 
yet  it  seems  to  me  that  the  consequent  uncritical  rejection  of  the 
legitimate  actual  infinite  is  no  lesser  violation  of  the  nature  of  things, 
which  must  be  taken  as  they  are."  ^ 

In  1904  Charles  Emile  Picard  of  Paris  expressed  himself  thus:^ 
"Since  the  concept  of  number  has  been  sifted,  in  it  have  been  found 
unfathomable  depths;  thus,  it  is  a  question  still  pending  to  know,  be- 
tween the  two  forms,  the  cardinal  number  and  the  ordinal  number, 
under  which  the  idea  of  number  presents  itself,  which  of  the  two  is 
anterior  to  the  other,  that  is  to  say,  whether  the  idea  of  number  prop 
erly  so  called  is  anterior  to  that  of  order,  or  if  it  is  the  inverse.  It 
seems  that  the  geometer-logician  neglects  too  much  in  these  questions 
psychology  and  the  lessons  uncivilized  races  give  us;  it  would  seem  to 
result  from  these  studies  that  the  priority  is  with  the  cardinal  number." 

Applied  Mathematics.    Celestial  Mechanics 

Notwithstanding  the  beautitul  developments  of  celestial  mechanics 
reached  by  P.  S.  Laplace  at  the  close  of  the  eighteenth  century,  there 
was  made  a  discovery  on  the  first  day  of  the  nineteenth  century  which 
])rcsented  a  problem  seemingly  beyond  the  power  of  that  analysis. 
We  refer  to  the  discovery  of  Ceres  by  Giuse])pe  Piazzi  in  Italy,  which 
became  known  in  Germany  just  after  the  philosopher  G.  W.  F.  Hegel 
had  published  a  dissertation  ])ro\'ing  a  priori  that  such  a  discovery 
could  not  be  made.  From  the  positions  of  the  planet  observed  by 
Piazzi  its  orbit  could  not  be  satisfactorily  calculated  by  the  old 
methods,  and  it  remained  for  the  genius  of  K.  F.  Gauss  to  devise  a 
method  of  calculating  elliptic  orbits  which  was  free  from  the  assumption 
of  a  small  eccentricity  and  inclination.  Gauss'  method  was  develojied 
further  in  his  Theoria  Motus.  The  new  planet  was  re-discovered  with 
aid  of  Gauss'  data  by  II.  W.  M.  Olbers,  an  astronomer  who  promoted 
science  not  only  by  his  own  astronomical  studies,  but  also  by  discern- 

'  See  F.  Cajori,  "History  of  Zeno  s  Arguments  on  Motion,"  Am.  Math.  Monthly, 
Vol.  22,  1915,  I).  114. 

-  G.  Cantor,  Ziim  Proi)lcm  cics  rictualen  Uncndlichcn,  Nalur  uitd  Ojjcnbarung, 
Bd.  32,  1886,  p.  226;  quoted  from  Moritz,  Manorabitia  malficmalica,  1914,  p.  .^7 

'  Congress  of  Arts  aiui  Science,  St.  LouiSj  1904,  Vol.  I,  p.  408. 


448  A  HISTORY  OF  MATHEMATICS 

ing  and  directing  towards  astronomical  pursuits  the  genius  of  F.  W, 
Bessel. 

Friedrich  Wilhelm  Bessel  ^  (1784-ICS46)  was  a  native  of  Minden  in 
Westphalia.  Fondness  for  figures,  and  a  distaste  for  Latin  grammar 
led  him  to  the  choice  of  a  mercantile  career.  In  his  fifteenth  year  he 
became  an  apprenticed  clerk  in  Bremen,  and  for  nearly  seven  years 
he  devoted  his  days  to  mastering  the  details  of  his  business,  and  part 
of  his  nights  to  study.  Hoping  some  day  to  become  a  supercargo  on 
trading  exi^editions,  he  became  interested  in  observations  at  sea. 
With  a  sextant  constructed  by  him  and  an  ordinary  clock  he  deter- 
mined the  latitude  of  Bremen.  His  success  in  this  inspired  him  for 
astronomical  study.  One  work  after  another  was  mastered  by  him, 
unaided,  during  the  hours  snatched  from  sleep.  From  old  observa- 
tions he  calculated  the  orbit  of  Halley's  comet.  Bessel  introduced 
himself  to  H.  W.  M.  Olbers,  and  submitted  to  him  the  calculation, 
which  Olbers  imniediately  sent  for  publication.  Encouraged  by  Ol- 
bers, Bessel  tunied  his  back  to  the  prospect  of  affluence,  chose  poverty 
and  the  stars,  and  became  assistant  in  J.  H.  Schroter's  observatory  at 
Lilienthal.  Four  years  later  he  was  chosen  to  superintend  the  con- 
struction of  the  new  observatory  at  Konigsberg.^  In  the  absence  of 
an  adequate  mathematical  teaching  force,  Bessel  was  obliged  to  lecture 
on  mathematics  to  prepare  students  for  astronomy.  He  was  relieved 
of  this  work  in  1825  by  the  arrival  of  C.  0.  J.  Jacobi.  We  shall  not 
recount  the  labors  by  which  Bessel  earned  the  title  of  founder  of 
modem  practical  astronomy  and  geodesy.  As  an  observer  he  towered 
far  above  K.  F.  Gauss,  but  as  a  mathematician  he  reverently  bowed 
before  the  genius  of  his  great  contemporary.  Of  Bessel's  papers,  the 
one  of  greatest  mathematical  interest  is  an  "  U ntcrsuchung  des  Theils 
der  plandarischen  Storungcn,  wclchcr  aus  dcr  Bavegung  der  Sonne 
enlstehV'  (1824),  in  which  he  introduces  a  class  of  transcendental 
functions,  Jn(x),  much  used  in  applied  mathematics,  and  known  as 
"Bessel's  functions."  He  gave  their  principal  properties,  and  con- 
structed tables  for  their  evaluation.  It  has  been  observed  that  Bes- 
sel's functions  appear  much  earlier  in  mathematical  -literature.^  Such 
functions  of  the  zero  order  occur  in  papers  of  Daniel  BemouUi  (1732) 
and  L.  Eulcr  on  vibration  of  heavy  strings  suspended  from  one  end. 
All  of  Bessel's  functions  of  the  first  kind  and  of  integral  orders  occur 
in  a  paper  by  L.  Euler  (1764)  on  the  vibration  of  a  stretched  elastic 
membrane.  In  1878  Lord  Rayleigh  proved  that  Bessel's  functions 
are  merely  particular  cases  of  Laplace's  functions.  J.  W.  L.  Glaisher 
illustrates  by   Bessel's   functions   his   assertion   that   mathematical 

1  Bessel  als  Bremer  Handlungslchrlins^,  Bremen,  1800. 

*J.  Frantz,  Fcslrede  aus  Vcraulassuug  von  BcsscFs  htnderlj'dhrigcm  Gehnrtstag, 
Konigsl)er^,  1884. 

3  Miixiiiie  H6clier,  "A  bit  of  Mathemutiail  History,"  Bull,  of  Ihc  N.  Y.  Math.  Soc, 
\'ol.  II,  i8y3,  p.  107. 


APPLIED  MATHEMATICS  449 

1, ranches  growing  out  of  j)hysical  inquiries  as  a  rule  "lack  the  easy 
flow  or  homogeneity  of  form  wliich  is  characteristic  of  a  mathematical 
theory  ]iroperly  so  called."  These  functions  have  been  studied  by 
Carl  Theodor  Anger  (1S03-1S5S)  of  Danzig,  Oskar  Schlomilch  (1S23- 
]()oi)  of  Dresden  who  was  the  founder  in  1S56  of  the  Zeitschrift  fur 
Mathematik  mid  Physik,  R.  Lipschitz  of  Bonn,  Carl  Neumann  of 
Leipzig,  Eugen  Lommel  (1S37-1S99)  of  Munich,  Isaac  Todhunter  of  St. 
John's  College,  Cambridge. 

Prominent  among  the  successors  of  P.  S.  Laplace  are  the  follow- 
ing: Simeon  Denis  Poisson  (17S1-1840),  who  wrote  in  1808  a  classic 
Memoire  sur  les  inegalitcs  scculaires  des  moyens  mouvements  des  plan- 
etcs.  Giovanni  Antonio  Amaedo  Plana  (17S1-1S64)  of  Turin,  a  nephew 
of  J.  Lagrange,  who  published  in  iSii  a  Mcmoria  sulla  teoria  dclV 
attrazione  dcgli  sjcroidi  ellilici,  and  contributed  to  the  theory  of  the 
moon.  Peter  Andreas  Hansen.  (1705-1S74)  of  Gotha,  at  one  time  a 
clockmaker  in  Tondern,  then  II.  C.  Shumacher's  assistant  at  Altona, 
and  finally  director  of  the  observatory  at  Gotha,  wrote  on  various 
astronomical  subjects,  but  mainly  on  the  lunar  theory,  which  he 
elaborated  in  his  work  Fundamenta  nova  invest igationes  orbitce  verce  quam 
Luna  pcrlustrat  (1838),  and  in  subsequent  investigations  embracing 
extensive  lunar  tables.  George  Biddel  Airy  (1801-1S92),  royal  as- 
tronomer at  Greenwich,  ])ublishcd  in  1S26  his  Mathematical  Tracts 
on  the  Lunar  and  Planetary  Theories.  These  researches  were  later 
greatly  extended  by  him.  August  Ferdinand  Mobius  (1790-1868) 
of  Leipzig  \\Tote,  in  1842,  Elcmcnte  dcr  Mcchanik  des  Himmels. 
Urbain  jean  Joseph  Leverrier  (1811-1S77)  of  Paris  wrote,  the 
Rccherches  Aslronomiques,  constituting  in  part  a  new  elaboration  of 
celestial  mechanics,  and  is  famous  for  his  theoretical  discovery  of 
Neptune.  John  Couch  Adams  (1S19-1S92)  of  Cambridge  divided 
with  Leverrier  the  honor  of  the  mathematical  discovery  of  Nep- 
tune, and  pointed  out  in  1853  that  Laplace's  explanation  of  the 
secular  acceleration  of  the  moon's  mean  motion  accounted  for  only 
half  the  observed  acceleration.  Charles  Eugene  Delaunay  (bom 
1S16,  and  dro^\^^ed  off  Cherbourg  in  1872),  professor  of  mechanics  at 
the  Sorbonne  in  Paris,  explained  most  of  the  remaining  acceleration  of 
the  moon,  unaccounted  for  by  Laplace's  theory  as  corrected  by  J.  C. 
Adams,  by  tracing  the  cITect  of  tidal  friction,  a  theory  previously 
suggested  independently  by  Immanuel  Kant,  Robert  Mayer,  and 
William  Fcrrcl  of  Kentucky.  G.  H.  Darwin  of  Cambridge  made 
some  very  remarkable  investigations  on  tidal  friction. 

Sir  George  Howard  Darwin  (1845-1912),  a  son  of  the  naturalist 
Cliarles  Darwin,  entered  Trinity  College,  Cambridge,  was  Second 
Wrangler  in  186S,  Lord  Moulton  being  Senior  Wrangler.  He  began 
in  1S75  to  publish  important  pajiers  on  the  aiii)lication  of  the  theory 
of  tidal  friction  to  the  evolution  of  the  solar  system.  The  earth-moon 
Tiv'stem  was  found  to  form  a  unique  exam[^le  within  the  solar  system 


450  A  HISTORY  OF  MATHEMATICS 

of  its  particular  mode  of  evolution.  He  traced  back  the  changes  m 
the  figures  of  the  earth  and  moon,  until  they  united  into  one  pear- 
shaped  mass.  This  theory  received  confirmation  in  1SS5  from  a  paper 
in  Acia  math.,  Vol.  7  by  H.  Poincare  in  which  he  enunciates  the  prin- 
ciple of  exchange  of  stabilities.  H.  Poincare  and  Darwin  arrived  at 
the  same  pear-shaped  figure,  Poincare  tracing  the  process  of  evolution 
forwards,  Darwin  proceeding  backwards  in  time.  Questions  of 
stability  of  this  changing  pear-shaped  figure  occupied  Darwin's  later 
years.  Researches  along  the  same  line  were  made  by  one  of  his 
))upils,  James  H.  Jeans  of  Trinity  College,  Cambridge. 

About  the  same  time  that  George  Danvin  began  his  researches, 
George  William  Hill  (1S3S-1914)  of  the  Nautical  Almanac  Office 
in  Washington  began  to  study  the  moon.  Hill  was  born  at  Nyack, 
New  York,  graduated  at  Rutgers  College  in  1859,  and  was  an  as- 
sistant in  the  Nautical  Almanac  Office  from  his  graduation  till  1892, 
.vhen  he  resigned  to  pursue  further  the  original  researches  which 
brought  him  distinction.  In  1877  he  published  Researches  on  Lunar 
Theory,  in  which  he  discarded  the  usual  mode  of  procedure  in  the 
])roblem  of  three  bodies,  by  which  the  problem  is  an  extension  of  the 
case  of  two  bodies.  Following  a  suggestion  of  Eulcr,  Hill  takes  the 
earth  finite,  the  sun  of  infinite  mass  at  an  infinite  distance,  the  moon 
infinitesimal  and  at  a  finite  distance.  The  differential  equations  which 
express  the  motion  of  the  moon  under  the  limitations  adopted  are 
fairly  simple^  and  practically  useful.  "It  is  this  idea  of  Hill's  that 
has  so  profoundly  changed  the  wiiole  outlook  of  celestial  mechanics. 
H.  Poincare  took  it  up  as  the  basis  of  his  celebrated  prize  essay  of 
1887  on  the  problem  of  three  bodies  and  afterwards  expanded  his 
work  into  the  three  volumes,  Les  mclhodes  nouvelles  de  la  mccanique 
celeste,^'  1S92-1899.  It  seems  that  at  first  G.  H.  Darwin  paid  little 
attention  to  Hill's  paper;  Darwin  often  spoke  of  his  difficulties  in 
assimilating  the  work  of  others.  However  in  18S8  he  recommended 
to  E.  W.  Brown,  now  professor  at  Yale,  the  study  of  Hill.  Nor  docs 
Darwin  seem  to  have  studied  closely  the  "planetesimal  hy]:)othesis " 
of  T.  C.  Chamberlin  and  F.  R.  Moulton  of  the  University  of  Chicago. 
A  marked  contrast  between  G.  H.  Darwin  and  H.  Poincare  lay  in 
the  fact  that  Darwin  did  not  undertake  investigations  for  their 
mathematical  interest  alone,  while  H.  Poincare  and  some  of  his 
followers  in  applied  mathematics  "have  less  interest  in  the  phenomena 
than  in  the  mathematical  processes  which  are  used  by  the  student 
of  the  i^henomena.  They  do  not  expect  to  examine  or  predict  physical 
events  but  rather  to  take  up  the  special  classes  of  functions,  differen- 
tial equations  or  series  which  have  been  used  by  astronomers  or  phy- 
sicists, to  examine  their  i)roi:)erties,  the  validity  of  the  arguments  and 
the  limitations  which  must  be  placed  01:  the  results"  (E.  W.  Brown). 

'  We  are  iisins?  E.  W.  Brown's  article  in  Scientific  Papers  by  Sir  G.  H.  Darwin^ 
Vol.  V,  lyiO,  pp.  .\.\.\iv-lv. 


APPLIED  MATHEMATICS  451 

Prominent  in  mathematical  astronomy  was  Simon  Newcomb 
(1835-1909),  the  son  of  a  country  school  teacher.  He  was  born  at  Wal- 
lace in  Nova  Scotia.  Althou|2;h  he  attended  for  a  year  the  Lawrence 
Scientific  School  at  Harvard  University,  he  was  essentially  self-taught. 
In  Cambridge  he  came  in  contact  with  B.  Pcirce,  B.  A.  Gould,  J.  D. 
Runkle,  and  T.  H.  SalTord.  In  1861  he  was  a]:)pointed  professor  in 
the  United  States  Navy;  in  1877  he  became  superintendent  of  the 
American  Ephemeris  and  Nautical  Almanac  Office.  This  position 
he  held  for  twenty  years.  During  1S84-1S95  he  was  also  professor 
of  mathematics  and  astronomy  at  the  Johns  Ho])kins  University, 
and  editor  of  the  American  Journal  of  Alalhemalics.  His  researches 
were  mainly  in  the  astronomy  of  position,  in  which  line  he  was  pre- 
eminent. In  the  comparison  between  theory  and  observation,  in 
deducing  from  large  masses  of  observations  the  results  which  he 
needed  and  which  would  form  a  basis  of  comparison  with  theory, 
he  was  a  master.  As  a  supplement  to  the  Nautical  Almanac  for  1897 
he  published  the  Elements  of  the  Four  Inner  planets,  and  the  Funda- 
mental Constants  of  Astronomy,  which  gathers  together  Newcomb's 
life-work.^  For  the  unravelling  of  the  motions  of  Jupiter  and  Saturn, 
S.  Newcomb  enlisted  the  services  of  G.  W.  Hill.  All  the  publications 
of  the  tables  of  the  planets,  except  those  of  Jupiter  and  Saturn,  bear 
Newcomb's  name.  These  tables  sui)plant  those  of  Leverrier.  S.  New- 
comb devoted  much  time  to  the  moon.  He  investigated  the  errors  in 
Hansen's  lunar  tables  and  continued  the  lunar  researches  of  C.  E. 
Delaunay.  Brief  reference  has  already  been  made  to  G.  W.  Hill's 
lunar  work  and  his  contribution  of  an  elegant  paper  on  certain  possible 
abbreviations  in  the  computation  of  the  long-period  of  the  moon's 
motion  due  to  the  direct  action  of  the  planets,  and  made  elaborate 
determination  of  the  inequalities  of  the  m.oon's  motion  due  to  the 
figure  of  the  earth.  He  also  computed  certain  lunar  inequalities  due 
to  the  action  of  Jupiter. 

The  mathematical  discussion  of  Saturn's  rings  was  taken  up  first 
by  P.  S.  Laplace,  who  demonstrated  that  a  homogeneous  solid  ring 
could  not  be  in  equilibrium,  and  in  1851  by  B.  Peirce,  who  [:)roved 
their  non-solidity  by  showing  that  even  an  irregular  solid  ring  could 
not  be  in  equilibrium  about  Saturn.  The  mechanism  of  these  rings 
was  investigated  by  James  Clerk  Maxwell  in  an  essay  to  which  the 
Adams  prize  was  awarded.  He  concluded  that  they  consisted  of  an 
aggregate  of  unconnected  particles.  "Thus  an  idea  put  forward  as  a 
speculation  in  the  seventeenth  century,  and  afterwards  in  the  eight- 
eenth century  by  J.  Cassini  and  Thomas  Wright,  was  mathematically 
demonstrated  as  the  only  possible  solution."  ^ 

The  progress  in  methods  of  computing  planetary,  asteroidal,  and 
cometary  orbits  has  proceeded  along  two  more  or  less  distinct  lines, 

*  E.  W.  Brown  in  Bull.  Am.  Math.  Soc,  Vol.  t6,  igio,  p.  353. 

*  VV.  W.  Bryant,  A  Uislory  of  Astronomy,  London,  1907,  p.  233. 


452  A  HISTORY  OF  MATHEMATICS 

the  one  marked  out  by  P.  S.  La])]ace,  the  other  by  K.  F.  Gaiiss.^  La- 
place's method  jiossessed  theoretical  advantages,  but  lacked  practical 
applicability  for  the  reason  that  in  the  second  approximation  the 
results  of  the  first  ajiproximation  could  be  used  only  in  i^art  and  the 
computation  had  to  be  gone  over  largely  anew.  To  avoid  this  labor 
in  finding  asteroidal  and  cometary  orbits,  Heinrich  W.  M.  Gibers 
(1758- 1840)  and  K.  F.  Gauss  devised  more  expeditious  processes  for 
carrying  out  the  second  approximation.  The  Gaussian  procedure 
was  refined  and  simplified  by  Johann  Franz  Encke  (1791-1865), 
Francesco  Carlini  (178.^-1862),  F.  W.  Bessel,  P.  A.  Hansen,  and  es- 
pecially by  Theodor  von  Oppolzer  (1841-1S86)  of  Vienna  whose 
method  has  been  used  by  practical  astronomers  down  to  the  present 
day.  Most  original  among  the  new  elaborations  of  Gauss'  method  is 
that  of  J.  Willard  Gibbs  of  Yale,  which  employs  vector  analysis  and, 
though  rather  complicated,  yields  remarkable  accuracy  even  in  the 
first  approximation.  Gibbs'  procedure  was  modified  in  1905  by  J. 
Frischauf  of  Graz.  P.  S.  Laplace's  method  has  attracted  mathemati- 
cians by  its  elegance.  It  received  the  attention  of  A.  L.  Cauchy, 
Antoine  Yvon  Villarceau  (1813-1883)  of  the  Paris  Observatory, 
Rodolphe  Radau  of  Paris,  II.  Bruns  of  Leipzig,  and  H.  Poincare.  Paul 
Harzer  of  Kiel  and  especially  Armin  Otto  Leuschner  of  the  University 
of  California  made  striking  advances  in  rendering  Laplace's  method 
available  for  rapid  computation.  Leuschner  adopts  from  the  start 
geocentric  co-ordinates  and  considers  the  effects  of  the  perturbating 
body  in  the  very  first  approximation;  it  is  equally  applicable  to  plane- 
tary and  to  cometary  orbits.- 

Problem  of  Three  Bodies 

The  problem  of  three  bodies  has  been  treated  in  various  ways  since 
the  time  of  J.  Lagrange,  and  some  decided  advance  towards  a  more 
complete  solution  has  been  made.  Lagrange's  particular  solution 
based  on  the  constancy  of  the  relative  distances  of  the  three  bodies, 
one  from  the  other  (called  by  L.  0.  Hesse  the  reduced  problem  of 
three  bodies)  has  recently  been  modified  by  Carl  L.  Charlier  of  the 
observatory  at  Lund,  in  which  the  mutual  distances  are  replaced 
by  the  distances  from  the  centre  of  gravity.^  This  new  form  possesses 
no  marked  advantage,  "Theoretical  interest  in  the  Lagrangian  solu- 
tions has  been  increased,"  says  E.  O.  Lovett,  "by  K.  F.  Sundman's 
theorem  that  the  more  nearly  all  three  bodies  in  the  general  ])roblem 
tend  to  collide  simultaneously,  the  more  nearly  do  they  tend  to  as- 
sume one  or  the  other  of  Lagrange's  configurations;  .  .  .  practical 

^  We  are  usinf]C  an  article  by  A.  Venturi  in  Rivista  di  Astronomia,  June,  IQ13. 

^  For  a  fuller  historical  account,  sec  A.  O.  Leuschner  in  Science,  N.  S.,  Vol.  45, 
1917,  pp.  571-584- 

3  We  are  drawing  from  E.  O.  Lovett's  "The  Problem  of  Three  Bodies"  in  Sciena. 
N.  S.,  Vol.  29,  1909,  pp.  81-91. 


APPLIED  MATHEMATICS  45:5 

interest  in  them  has  been  revived  by  the  discovery  of  three  small 
planets,  1906  T.  G.,  1906  V.  Y.,  1907  X.  M.,  near  the  equilateral  tri- 
angular points  of  the  Sun-Jui)iter-Asteroid  system.  ...  R.  Leh- 
mann-Filhes,  R.  Hoppc,  and  Otto  Dziobek,  all  three  of  Berlin,  have 
generalized  the  exact  solutions  to  cases  of  more  than  three  bodies 
{ihiced  on  a  line  or  at  the  vertices  of  a  regular  polygon  or  polyhe- 
dron. .  .  .  Among  the  most  interesting  extensions  of  Lagrange's 
theorem  are  those  due  to  T.  Banachie\'itz  of  Kasan  and  F.  R.  Moul- 
ton."  In  191 2  H.  Poincare  indicated  that  on  the  basis  of  a  ring 
representation  (but  in  Keplerian  variables),  that  if  a  certain  geometric 
theorem  (later  established  by  G.  D.  Birkhoff  of  Harvard  University) 
were  true,  the  existence  of  an  infinite  number  of  periodic  solutions 
would  follow  in  the  restricted  problem  of  three  bodies.  These  results 
were  amplified  by  G.  D.  Birkhoff.^  The  so-called  isosceles- triangle 
solutions  of  the  problem  of  three  bodies  (periodic  solutions  in  which 
two  of  the  masses  are  finite  and  equal,  while  the  third  body  moves  in 
a  straight  line  and  remains  equidistant  from  the  equal  bodies)  received 
the  attention  of  Giulio  Pavanini  of  Treviso  in  Italy  in  1907,  W.  D. 
MacMillan  of  Chicago  in  191 1,  and  D.  Buchanan  of  Ontario  in  1914. 
G.  W.  Hill,  in  his  researches  on  lunar  theory,  added  in  1877  to  the 
Lagrangian  periodic  solution,  which  for  105  years  had  been  the  only 
such  solution  known,  another  periodic  solution  which  could  serve  as 
a  starting  ])oint  for  a  study  of  the  moon's  orbit.  Says  E.  O.  Lovett: 
"With  these  memoirs  he  broke  ground  for  the  erection  of  the  ne^ 
science  of  dynamical  astronomy  whose  mathematical  foundations 
were  laid  broad  and  deep  by  Poincare,"  in  a  research  which  in  1889 
won  a  prize  offered  by  King  Oscar  II,  and  which  he  developed  more 
fully  later.  The  original  memoir  of  Poincare,  says  Moulton,  "con- 
tained an  error  which  was  discovered  by  E.  Plu"agmen,  of  Stockholm, 
but  it  affected  only  the  discussion  of  the  existence  of  the  asymptotic 
solutions;  and  in  correcting  this  part  H.  Poincare  .  .  .  confessed 
fully  his  obligations  to  Phragmen.  .  .  .  There  is  not  the  slightest 
doubt  that  in  spite  of  it  .  .  .  the  prize  was  correctly  bestowed." 
llie  researches  of  G.  W.  Hill  and  H.  Poincare  have  been  continued 
mainly  by  E.  W.  Brown,  G.  H.  Darwin,  F.  R.  Moulton,  Hugo  Gylden 
(1S41-1896)  of  Stockholm,  P.  Painleve,  C.  L.  W.  Charlier,  S.  E. 
Stromgren,  and  T.  Levi-Civita,  in  which  questions  of  stabiUty  have  re- 
ceived much  attention.  The  general  question,  whether  the  solar 
system  is  stable,  was  affirmed  by  eighteenth  century  mathematicians; 
it  was  re-opened  by  K.  Weierstrass  who,  in  the  last  years  of  his  life, 
devoted  considerable  attention  to  it.  Expressions  for  the  co-ordinates 
of  the  planets  converge  either  not  at  all  or  for  only  limited  time.  In 
addition  to  the  complex  mixture  of  known  cyclical  changes,  there 
niight,  perhaps,  be  a  small  residue  of  change  of  such  a  nature  that 
the  system  will  ultimately  be  wrecked.  At  present  no  rigorous  answer 
^  Bull.  Am.  Malh.  Soc,  Vol.  20,  1914,  p.  292. 


454  A  HISTORY  OF  MATHEMATICS 

has  been  given,  l)ut  "Poincarc  showed  that  solutions  exist  in  which 
the  motion  is  purely  periodic,  and  therefore  that  in  them  at  least  no 
disaster  of  collision  or  indefinite  departure  from  the  central  mass  will 
ever  occur"  (F.  R.  Moulton).  A  startling  result  was  Poincare's  dis- 
covery that  some  of  the  series  which  have  been  used  to  calculate  the 
positions  of  the  bodies  of  the  solar  system  are  divergent.  An  exam- 
ination of  the  reasons  why  the  divergent  series  gave  sufficiently  ac- 
curate results  gave  rise  to  the  theory  of  asymptotic  series  now  applied 
to  the  representation  of  many  functions.  Does  the  ultimate  diver- 
gence of  the  series  throw  doul:)t  upon  the  stability  of  the  solar  system? 
H.  Gylden  thought  that  he  had  overcome  the  difficulty,  but  H.  Poin- 
care  showed  that  in  part  it  still  exists.  Following  Poincare's  lead, 
E.  W.  Brown  has  formulated  the  sufficient  conditions  for  stability  in 
the  7Z-body  jiroblem.  T.  Levi-Civita  worked  out  criteria  in  which 
the  stability  is  made  to  depend  upon  that  of  a  certain  point  trans- 
formation associated  with  the  j^eriodic  function.  He  proved  the 
existence  of  zones  of  instability  surrounding  Jupiter's  orbit.  The 
new  methods  in  celestial  mechanics  have  been  found  useful  in  com- 
puting the  perturbations  of  certain  small  planets.  Material  advances 
in  the  problem  of  three  bodies  were  made  by  Karl  F.  Sundman  of 
Helsingfors  in  Finland,  in  a  memoir  which  received  a  prize  of  the 
Paris  Academy  in  1913.  This  research  is  along  the  path  first  blazed 
by  P.  Painleve,  continued  by  T.  Levi-Civita  and  others. 

In  the  transformation  and  reduction  of  the  three-body  problem,  "a 
principal  role  has  been  played  by  the  ten  known  integrals,  namely, 
the  six  integrals  of  motion  of  the  centre  of  gravity,  three  integrals  of 
angular  momentum,  and  the  integral  of  energy.  The  question  of 
further  progress  in  this  reduction  is  vitally  related  to  the  non-existence 
theorems  of  H.  Bruns,  H.  Poincare,  and  P.  Painleve.  H.  Bruns  demon- 
strated that  the  w-body  problem  admits  of  no  algebraical  integral 
other  than  the  ten  classic  ones,  and  H.  Poincare  jiroved  the  non- 
existence of  any  other  uniform  analytical  integral."  Other  researches 
3n  these  non-existence  theorems  are  due  to  P.  Painleve,  D.  A.  Grave, 
and  K.  Bohlin. 

E.  Picard  expresses  himself  as  follows:^  "What  admirable  recent 
researches  have  best  taught  them  [analysts]  is  the  immense  difficulty 
of  the  problem;  a  new  way  has,  however,  been  opened  by  the  study 
of  particular  solutions,  such  as  the  periodic  solutions  and  the  asymp- 
totic solution  which  have  already  been  utilized.  It  is  not  perhaps 
so  much  because  of  the  needs  of  practice  as  in  order  not  to  avow  it- 
self vanquished,  that  analysis  will  never  resign  itself  to  abandon,  with- 
out a  decisive  victory,  a  subject  where  it  has  met  so  many  brilliant 
triumphs;  and  again,  what  more  beautified  field  could  the  theories 
new-born  or  rejuvenated  of  the  modern  doctrine  of  functions  find, 
to  essay  their  forces,  than  this  classic  problem  of  71  bodies?" 
'•  Congress  of  AHs  and  Science,  St.  Louis,  1904,  Vol.  I,  p.  512. 


APPLIED  MATHEMATICS 


45: 


Among  valuable  text-books  on  mathematical  astronomy  of  the 
nineteenth  century  rank  the  following  works:  Manual  of  Spherical 
'jnd  Practical  Astronomy  by  William  Chauvencl  (1863),  Practical  and 
Spherical  Astronomy  by  Robert  Main  of  Cambridge,  Theoretical  As- 
tronomy by  James  C.  Watson  of  Ann  Arbor  (1868),  Traite  elcnentaire 
de  Mecanique  Celeste  of  H.  Resal  of  the  Ecole  Polytechnique  in  Paris, 
Cours  d'Aslronomie  de  V Ecole  Polytechnique  by  Faye,  Traite  de  Mecani- 
que Celeste  by  F.  F.  Tisserand,  Lehrbwh  dcr  Bahnbestimmuns,  by  T. 
bppolzer,  Mathematische  Thcorien  der  Phinclenbeivegung  by  O.  Dziobek, 
translated  into  English  by  M.  W.  Harrington  and  W.  J.  Hussey. 

General  Mechanics 

During  the  nineteenth  century  we  have  come  to  recognize  the  ad- 
vantages frequently  arising  from  a  geometrical  treatment  of  me- 
chanical problems.  To  L.  Poinsot,  M.  Chasles,  and  A.  F.  Mobius  we 
owe  the  most  important  developments  made  in  geometrical  mechanics. 
Louis  Poinsot  (i 777-1859),  a  graduate  of  the  Polytechnic  School  in 
Paris,  and  for  many  years  member  of  the  superior  council  of  public 
instruction,  published  in  1804  his  Elements  de  Statique.  This  work 
is  remarkable  not  only  as  being  the  earhest  introduction  to  synthetic 
mechanics,  but  also  as  containing  for  the  first  time  the  idea  of  couples, 
which  was  applied  by  Poinsot  in  a  publication  of  1834  to  the  theory 
of  rotation.  A  clear  conception  of  the  nature  of  rotary  motion  was 
conveyed  by  Poinsot's  elegant  geometrical  representation  by  means 
of  an  ellipsoid  rolling  on  a  certain  fixed  plane.  This  construction  was 
extended  by  J.  J.  Sylvester  so  as  to  measure  the  rate  of  rotation  of  the 
ellipsoid  on  the  plane. 

A  particular  class  of  dynamical  problems  has  recently  been  treated 
geometrically  by  Sir  Robert  Stawell  Ball  (1840-19 13)  at  one  time 
astronomer  royal  of  Ireland,  later  Lowndean  Professor  of  Astronomy 
and  Geometry  at  Cambridge.  His  method  is  given  in  a  work  entitled 
Theory  of  Screws,  Dublin,  1876,  and  in  subsequent  articles.  Modern 
geometry  is  here  drawn  upon,  as  was  done  also  by  W.  K.  ClitTord 
in  the  related  subject  of  Bi-quaternions.  Arthur  Buchheim  (1859- 
1SS8),  of  Manchester  showed  that  H.  G.  Grassmann's  Ausdehnungs- 
lehre  supplies  all  the  necessary  materials  for  a  simple  calculus  of  screws 
in  elliptic  space.  Horace  Lamb  applied  the  theory  of  screws  to  the 
question  of  the  steady  motion  of  any  solid  in  a  fluid. 

Advances  in  theoretical  mechanics,  bearing  on  the  integration  and 
the  alteration  in  form  of  dynamical  equations,  were  made  since  J. 
Lagrange  by  S.  D.  Poisson,  Sir  William  Rowan  Hamilton,  C.  G.  J. 
Jacobi,  Madame  Koalcvski,  and  others.  J.  Lagrange  had  established 
the  "Lagrangian  form"  of  the  equations  of  motion.  He  had  given  a 
cheory  of  the  variation  of  the  arbitrary  constants  which,  however, 
turned  out  to  be  less  fruitful  in  rt^sults  than  a  theorv  advanced  by  S.  D. 


456  A  HISTORY  OF  MATHEMATICS 

Poisson.^  Poisson's  theory  of  the  variation  of  the  arbitrary  constants 
and  the  method  of  integration  thereby  afforded  marked  the  first 
onward  step  since  J.  Lagrange.  Then  came  the  researches  of  Sir 
William  Rowan  Hamilton.  His  discovery  that  the  integration  of  the 
dynamic  differential  equations  is  connected  with  the  integration  of  a 
certain  partial  differential  equation  of  the  first  order  and  second  degree, 
grew  out  of  an  attempt  to  deduce,  by  the  undulatory  theory,  results 
in  geometrical  optics  previously  based  on  the  conceptions  of  the  emis- 
sion theory.  The  Philosophical  Transactions  of  1S33  and  1834  contain 
Hamilton's  papers,  in  which  appear  the  first  applications  to  mechanics 
of  the  principle  of  varying  action  and  the  characteristic  function, 
established  by  him  some  years  previously.  The  object  which  Hamilton 
proposed  to  himself  is  indicated  by  the  title  of  his  first  paper,  viz., 
the  discovery  of  a  function  by  means  of  which  all  integral  equations 
can  be  actually  represented.  The  new  form  obtained  by  him  for  the 
equation  of  motion  is  a  result  of  no  less  importance  than  that  which 
was  the  professed  object  of  the  memoir.  Hamilton's  method  of  in- 
tegration was  freed  by  C.  G.  J.  Jacobi  of  an  unnecessary  complica- 
tion, and  was  then  applied  by  him  to  the  determination  of  a  geodetic 
line  on  the  general  ellipsoid.  With  aid  of  elliptic  co-ordinates  Jacobi 
integrated  the  partial  differential  equation  and  expressed  the  equation 
of  the  geodetic  in  form  of  a  relation  between  two  Abelian  integrals. 
C.  G.  J.  Jacobi  applied  to  differential  equations  of  dynamics  the  theory 
of  the  ultimate  multiplier.  The  differential  equations  of  dynamics  are 
only  one  of  the  classes  of  differential  equations  considered  by  Jacobi. 
Dynamic  investigations  along  the  lines  of  J.  Lagrange,  Hamilton,  and 
Jacobi  were  made  by  J.  Liouville,  Adolphe  Desboves,  (1818-1888) 
of  Amiens,  Serret,  J.  C.  F.  Sturm,  Michel  Oscrogradski,  J.  Bertrand, 
William  Fishburn  Donkin  (1814-1S69)  cf  Oxford,  F.  Brioschi,  leading 
up  to  the  development  of  the  theory  of  a  system  of  canonical  integrals. 

An  important  addition  to  the  theory  of  the  motion  of  a  solid  body 
about  a  fixed  point  was  made  by  INladame  Sophie  Kovalevski  (1850- 
1891),  who  discovered  a  new  case  in  which  the  differential  equations 
of  motion  can  be  integrated.  By  the  use  of  theta-functions  of  two 
independent  variables  she  furnished  a  remarkable  example  of  how 
the  modern  theory  of  functions  may  become  useful  in  mechanical 
problems.  She  was  a  native  of  Moscow,  studied  under  K.  Weierstrass, 
obtained  the  doctor's  degree  at  Gottingen,  and  from  1884  until  her 
death  was  professor  of  higher  mathematics  at  the  University  of  Stock- 
holm. The  research  above  mentioned  received  the  Bordin  prize 
of  the  French  Academy  in  1888,  which  was  doubled  on  account  of 
the  exceptional  merit  of  the  paper. 

There  are  in  vogue  three  forms  for  the  expression  of  the  kinetir 
energy  of  a  dynamical  system:  the  Lagrangian,  the  Hamiltonian,  and 

'Arthur  Cayley,  "Report  on  the  Recent  Progress  of  Theoretical  Dynamicb," 
Report  Brr  ish  Ass'n  for  1857,  p.  7. 


APrLIED  MATHExMATICS  457 

a  modified  form  of  Lagrange's  equations  in  which  certain  velocities 
are  omitted.  The  kinetic  energy  is  expressed  in  the  first  form  as  a 
homogeneous  quadratic  function  of  the  velocities,  which  are  the  time- 
variations  of  the  co-ordinates  of  the  system;  in  the  second  form,  as 
a  homogeneous  quadratic  function  of  the  momenta  of  the  system; 
the  third  form,  elaborated  recently  by  Edward  John  Routh  (1831- 
1907)  of  Cambridge,  in  connection  with  his  theory  of  ''ignoration  of 
co-ordinates,"  and  by  A.  E.  Basset  of  Cambridge,  is  of  importance  in 
hydro-dynamical  problems  relating  to  the  motion  of  perforated  solids 
in  a  liquid,  and  in  other  branches  of  physics. 

Practical  importance  has  come  to  be  attached  to  the  principle  of 
mechanical  sim  litude.  By  it  one  can  determine  from  the  performance 
of  a  model  the  action  of  the  machine  constructed  on  a  larger  scale. 
The  principle  was  first  enunciated  by  I.  Newton  {Principia,  Bk.  11, 
Sec.  VIII,  Prop.  32),  and  was  derived  by  Joseph  Bertrand  from  the 
principle  of  virtual  velocities.  A  corollary  to  it,  applied  in  ship- 
building, is  named  after  the  British  naval  architect  William  Froude 
(1810-1879),  but  was  enunciated  also  by  the  French  engineer  Frederic 
Reech. 

The  present  problems  of  dynamics  differ  materially  from  those  of 
the  last  century.  The  explanation  of  the  orbital  and  axial  motions 
of  the  heavenly  bodies  by  the  law  of  universal  gravitation  was  the 
great  problem  solved  by  A.  C.  Clairaut,  L.  Euler,  D'Alembert,  J. 
Lagrange,  and  P.  S.  Laplace.  It  did  not  involve  the  consideration  of 
frictional  resistances.  In  the  present  time  the  aid  of  d^mamics  has 
been  invoked  by  the  physical  sciences.  The  problems  there  arising 
are  often  complicated  by  the  presence  of  friction.  Unlike  astronomical 
problems  of  a  century  ago,  they  refer  to  phenomena  of  matter  and 
motion  that  are  usually  concealed  from  direct  observation.  The  great 
pioneer  in  such  problems  is  Lord  Kelvin.  Wliile  yet  an  undergraduate 
at  Cambridge,  during  holidays  spent  at  the  seaside,  he  entered  upon 
researches  of  this  kind  by  working  out  the  theory  of  spinning  tops, 
which  previously  had  been  only  partially  explained  by  John  Hewitt 
Jellett  (1S17-18S8)  of  Trinity  College,  Dublin,  in  his  Treatise  on  the 
Theory  of  Friction  (1872),  and  by  Archibald  Smith  (1813-1872). 

Among  standard  works  on  mechanics  of  the  nineteenth  century  are 
C.  G.  J.  Jacobi's  Vorlesungen  uber  Dynamik,  edited  by  R.  F.  A.Clebsch, 
1866;  G.  R.  Kirchhojf's  Vorlesungen  iiber  mathemalische  Physik,  1876; 
Benjamin  Peirce's  Analytic  Mechanics,  1855;  /.  /.  Somoffs  Theoretische 
Mechanik,  1879;  P.  G.  Tait  and  W.  J.  Steele's  Dynamics  of  a  Particle, 
1856;  George  Minchin's  Treatise  on  Statics;  E.  J.  Routh' s  Dynamics  of 
a  System  of  Rigid  Bodies;  J .  C.  F.  Sturm's  Cours  de  Mecanique  del'Ecole 
Polytechnique.  George  M.  Minchin  (1845-1914)  was  professor  at  the 
Indian  engineering  college. 

In  1898  Felix  Klein  pointed  out  the  separation  which  existed  be- 
tween British  and  Continental  mathematical  research,  as  seen,  foi 


45S  A  HISTORY  OF  MATHEMATICS 

instance,  by  the  contents  of  E.  J.  Routh's  Dynamics,  which  contains 
the  results  of  twenty  years  of  research  along  that  hne  in  England  and, 
in  comparison  with  the  German  school,  emphasizes  a  concrete  and 
practical  treatment.  To  make  these  treasures  more  readily  accessible 
to  German  students,  Routh's  text  was  translated  into  German  by 
Adolf  Schepp  (1837-1905)  of  Wiesbaden  in  1898.  Particularly  strong 
was  Routh  in  the  treatment  of  small  oscillations  erf  systems;  the 
technique  of  integration  of  linear  differential  equations  with  con- 
stant coefficients  is  highly  developed,  except  that,  perhaps,  the  extent 
to  which  the  developments  are  valid  may  need  closer  examination. 
This  is  done  in  F.  Klein  and  A.  Sommerfeld's  Theorie  des  Kreisels, 
1897-1910.  This  last  work  gives  attention  to  the  theory  of  the  top, 
the  history  of  which  reaches  back  to  the  eighteenth  century. 

In  1744  Serson  started  on  a  ship  (that  was  lost),  to  test  the  prac- 
ticability of  the  artificial  horizon  furnished  by  the  poUshed  surface 
of  a  top.  This  idea  has  been  recently  revived  by  French  navigators.^ 
Serson's  top  induced  J.  A.  Segner  of  Halle  in  1755  to  give  precision 
to  the  theory  of  the  spinning  top,  which  was  taken  up  more  fully  by 
L.  Euler  in  1765  and  then  by  J.  Lagrange.  L.  Euler  considers  the 
motion  on  a  smooth  horizontal  plane.  Later  come  the  studies  due 
to  L.  Poinsot,  S.  D.  Poisson,  C.  G.  J.  Jacobi,  G.  R.  Kirchhoff,  Eduard 
Lottner  (1826-1887)  of  Lippstadt,  Wilhelm  Hess,  Clerk  Maxwell, 
E.  J.  Routh  and  finally  F.  Klein  and  A.  Sommerfeld.  In  1914  G. 
Greenhill  prepared  a  Report  on  Gyroscopic  Theory  ^  which  is  of  more 
direct  interest  to  engineers  than  is  Klein  and  Sommerfeld's  Theorie  des 
Kreisels,  developed  by  the  aid  of  the  theory  of  functions  of  a  complex 
variable.  Among  recenc  practical  applications  of  gyroscopic  action 
are  the  torpedo  exhibited  before  the  Royal  Society  of  London  in  1907 
by  Louis  Brennan,  also  Brennan's  monorail  system,  and  the  methods 
of  steadying  ships  and  aircraft,  devis^^d  by  the  American  engineer 
Elmer  A.  Sperry  and  by  Otto  Schlick  in  Germany. 

Among  the  deviations  of  a  projectile  from  the  theoretic  parabolic 
path  there  are  two  which  are  of  particular  interest.  One  is  a  shght 
bending  to  the  right,  in  the  northern  hemisphere,  owing  to  the  rotation 
of  the  earth;  it  was  explained  by  S.  D.  Poisson  (1838)  and  W.  Ferrel 
(1889).  The  other  is  due  to  the  rotation  of  the  projectile;  it  was  ob- 
served by  I.  Newton  in  tennis  balls  and  applied  by  him  to  explain 
certain  phenomena  in  his  corpuscular  theory  of  light;  it  was  known 
to  Benjamin  Robins  and  L.  Euler.  In  1794  the  Berhn  academy 
offered  a  prize  for  an  explanation  of  the  phenomenon,  but  no  satis- 
factory explanation  appeared  for  over  half  a  century.  S.  D.  Poisson 
in  1839  {Journ.  ecole  polyt.,  T.  27)  studied  the  effect  of  atmospheric 

>  See  A.  G.  Greenhill  in  Verhandl.  III.  Intern.  Congr.,  Heidelberg,  1^04,  Leipzig, 
1905,  p.  100.    We  are  summarizing  this  article. 

'^Advisory  CommiUee  for  Aeronautics,  Reports  and  Memoranda,  No.  146,  London, 
1914. 


APPLIED  MATHEMATICS 


459 


triction  against  the  rotating  sphere,  but  finally  admitted  that  friction 
was  not  sufficient  to  explain  the  deviations.  The  difference  in  the 
pressure  of  the  air  upon  the  rotating  sphere  also  demands  attention. 
An  explanation  on  this  basis,  which  was  generally  accepted  as  valid 
was  given  by  H.  G.  Magnus  (1802-1S70)  of  Berlin,  in  Poggendorff's 
Annalen,  Vol.  88,  1853.  In  connection  with  golf-balls  the  problem 
was  taken  up  by  Tait. 

Peter  Guthrie  Tait  (1831-1901)  was  born  at  Dalkeith,  studied  at 
Cambridge  and  came  out  Senior  Wrangler  in  1854,  which  was  a  sur- 
prise, as  W.  J.  Steele  had  been  generally  ahead  in  college  examinations. 
From  1854-1860  Tait  was  professor  of  mathematics  at  Belfast,  where 
he  studied  quaternions;  from  i860  to  his  death  he  held  the  chair  of 
Natural  Philosophy  at  Edinburgh.  Tait  found  the  problem  of  the 
flight  of  the  golf  ball  capable  of  exact  statement  and  approximate 
solution.  One  of  his  sons  had  become  a  brilliant  golfer.  Tait  at  first 
was  scoffed  at  when  he  began  to  offer  explanations  of  the  secret  of 
long  driving.  In  1SS7  {Nature,  36,  p.  502)  he  shows  that  "rotation" 
played  an  important  part,  as  estabUshed  experimentally  by  H.  G. 
Magnus  (1852).  Says  P.  G.  Tait:  "In  topping,  the  upper  part  of  the 
ball  is  made  to  move  forward  faster  than  does  the  center,  consequently 
the  front  of  the  ball  descends  in  virtue  of  the  rotation,  and  the  ball 
itself  skews  in  that  direction.  When  a  ball  is  undercut  it  gets  the 
opposite  spin  to  the  last,  and,  in  consequence,  it  tends  to  deviate  up- 
wards instead  of  downwards.  The  upward  tendency  often  makes  the 
path  of  a  ball  (for  a  part  of  its  course)  concave  upwards  in  spite  of 
the  effects  of  gravity.  .  .  ."  P.  G.  Tait  explained  the  influence  of 
the  underspin  in  prolonging  not  only  the  range  but  also  the  time  of 
flight.  The  essence  of  his  discovery  was  that  without  spin  a  ball 
could  not  combat  gravity  greatly,  but  that  with  spin  it  could  travel 
remarkable  distances.  He  was  fond  of  the  game  while  H.  Helmholtz 
(who  was  in  Scotland  in  187 1)  "could  see  no  fun  in  the  leetle  hole." 

P.  G.  Tait  generalized  in  1898  the  Josephus  problem  and  gave  the 
rule  for  n  persons,  certain  v  of  which  shall  be  left  after  each  m  ^^  man 
is  picked  out. 

The  deviations  of  a  body  falling  from  rest  near  the  surface  of  the 
earth  have  been  considered  in  many  memoirs  from  the  time  of  P.  S. 
Laplace  and  K.  F.  Gauss  to  the  present.  All  writers  agree  that  the 
body  will  deviate  to  the  eastward  with  respect  to  the  plumb-line  hung 
from  the  initial  point,  but  there  has  been  disagreement  regarding  the 
deviation  measured  along  the  meridian.  Laplace  found  no  meridional 
deviation,  Gauss  found  a  small  deviation  toward  the  equator.  Re- 
cently this  problem  has  commanded  the  attention  of  writers  in  the 
United  States.  R.  S.  Woodward,  president  of  the  Carnegie  Institution 
in  Washington,  found  in  1913  a  deviation  away  from  the  equator, 
F.  R.  Moulton  of  the  University  of  Chicago  found  in  19 14  a  formula 
indicating  a  southerly  deviation.     IF.  //.  Rover  of  Washington  Uni- 


46o  A  HISTORY  OF  MATHEMATICS 

versity  in  St.  Louis  has,  since  1901,  treated  the  subject  in  several 
articles  which  indicate  southerly  deviations.  He  declares  that  "no 
potential  function  is  known  that  fits  all  parts  of  the  earth,"  "that 
the  formula  of  Gauss,  the  three  formula?  of  Comte  de  Sparre  [Lyon, 
1905],  the  formula  of  Professor  F.  R.  Moulton,  and  my  first  formula, 
are  all  special  cases  of  my  general  formula."  ^ 

Fluid  Motion 

The  equations  which  constitute  the  foundation  of  the  theory  of 
fluid  motion  were  fully  laid  down  at  the  time  of  J.  Lagrange,  but  the 
solutions  actually  worked  out  were  few  and  mainly  of  the  irrotational 
type.  A  powerful  method  of  attacking  problems  in  fluid  motion  is 
that  of  images,  introduced  in  1843  by  G.  G.  Stokes  of  Pembroke  Col- 
lege, Cambridge.  It  received  little  attention  until  Sir  William  Thom- 
son's discovery  of  electrical  images,  whereupon  the  theory  was  ex- 
tended by  G.  G.  Stokes,  W.  M.  Hicks,  and  T.  C.  Lewis. 

George  Gabriel  Stokes  (18 19-1903)  was  born  at  Skreen,  County 
SUgo,  in  Ireland.  In  1837,  the  year  of  Queen  Victoria's  accession,  he 
commenced  residence  at  Cambridge,  where  he  was  to  find  his  home, 
almost  without  intermission,  for  sixty-six  years.  At  Pembroke  College 
his  mathematical  abilities  attracted  attention  and  in  184 1  he  graduated 
as  Senior  Wrangler  and  first  Smith's  prizeman.  He  distinguished 
himself  along  the  lines  of  applied  mathematics.  In  1845  he  published 
a  memoir  on  "Friction  of  Fluids  in  Motion."  The  general  motion  of 
a  medium  near  any  point  is  analyzed  into  three  constituents — a  mo- 
tion of  pure  translation,  one  of  pure  rotation  and  one  of  pure  strain. 
Similar  results  were  reached  by  H.  Helmholtz  twenty-three  years 
later.  In  applying  his  results  to  viscous  fluids,  Stokes  was  led  to 
general  dynamical  equations,  previously  reached  from  more  special 
hypotheses  by  L.  M.  H.  Navier  and  S.  D.  Poisson.  Both  Stokes  and 
G.  Green  were  followers  of  the  French  school  of  applied  mathemati- 
cians. Stokes  applies  his  equations  to  the  propagation  of  sound,  and 
shows  that  viscosity  makes  the  intensity  of  sound  diminish  as  the 
time  increases  and  the  velocity  less  than  it  would  otherwise  be — 
especially  for  high  notes.  He  considered  tne  two  elastic  constants  in 
the  equations  for  an  elastic  solid  to  be  independent  and  not  reducible 
to  one  as  is  the  case  in  Poisson's  theory.  Stokes'  position  was  sup- 
ported by  Lord  Kelvin  and  seems  now  generally  accepted.  In  1847 
Stokes  examined  anew  the  theory  of  oscillatory  waves.  Another 
paper  was  on  the  effect  of  internal  friction  of  fluids  on  the  motion  of 
pendulums.  He  assumed  that  the  viscosity  of  the  air  was  propor- 
tional to  the  density,  which  was  shown  later  by  Maxwell  to  be  erro- 
neous. In  1849  he  treated  the  ether  as  an  elastic  solid  in  the  study  of 
diffraction.    He  favored  Fresnel's  wave  theory  of  light  as  opposed  to 

'  See  Washinglon  University  Studies,  Vol.  ill,  191O,  op.  1.S3-1O8. 


APPLIED  .MATHP:MATICS  461 

fhe  corpuscular  theory  supported  by  David  Brewster.  In  a  report 
on  double  refraction  of  1S62  he  correlated  the  work  of  A.  L.  Cauchy, 
J.  MacCullagh,  and  G.  Green.  Assuming  that  the  elasticity  of  the 
ether  has  its  origin  in  deformation,  he  inferred  that  J.  MacCuUagh's 
theory  was  contrary  to  the  laws  of  mechanics,  but  recently  J.  Larmor 
has  shown  that  J.  Md.xJullagh's  equations  may  be  explained  on  the 
supposition  that  what  is  resisted  is  not  deformation,  but  rotation. 
Stokes  wrote  on  Fourier  series  and  the  discontinuity  of  arbitrary 
constants  in  semi-convergent  expansions  over  a  plane.  His  contribu- 
tions to  hydrodynamics  and  optics  are  fundamental.  In  1849  William 
Thomson  (Lord  Kelvin)  gave  the  maximum  and  minimum  theorem 
peculiar  to  hydrodynamics,  which  was  afterwards  extended  to  dynam- 
ical i-)roblems  in  general. 

A  new  epoch  in  the  progress  of  hydrodynamics  was  created,  in  1856, 
by  H.  Helmholtz,  who  worked  out  remarkable  properties  of  rotational 
motion  in  a  homogeneous,  incompressible  fluid,  devoid  of  viscosity. 
He  showed  that  the  vortex  filaments  in  such  a  medium  may  possess 
any  number  of  knottings  and  twistings,  but  are  either  endless  or  the 
ends  are  in  the  free  surface  of  the  medium ;  they  are  indivisible.  These 
results  suggested  to  William  Thomson  (Lord  Kelvin)  the  possibility 
of  founding  on  them  a  new  form  of  the  atomic  theory,  according  to 
which  every  atom  is  a  vortex  ring  in  a  non-frictional  ether,  and  as 
such  must  be  absolutely  permanent  in  substance  and  duration.  The 
vortex-atom  theory  was  discussed  by  J.  J.  Thomson  of  Cambridge 
(born  1856)  in  his  classical  treatise  on  the  Motion  of  Vortex  Rings,  to 
which  the  Adams  Prize  was  awarded  in  1882.  Papers  on  vortex  motion 
have  been  published  also  by  Horace  Lamb,  Thomas  Craig,  Henry  A. 
Rowland,  and  Charles  Chree  of  Kew  Observatory. 

The  subject  of  jets  was  investigated  by  H.  Helmholtz,  G.  R.  Kirch- 
hoff,  J.  Plateau,  and  Lord  Rayleigh;  the  motion  of  fluids  in  a  fluid  by 
G.  G.  Stokes,  W.  Thomson  (Lord  Kelvin),  H.  A.  Kopcke,  G.  Greenhill, 
and  H.  Lamb;  the  theory  of  viscous  fluids  by  H.  Navier,  S.  D.  Poisson, 
B.  de  Saint-Vcnant,  Stokes,  Oskar  Emil  Meyer  (1834-1909)  of  Breslau, 
A.  B.  Stefano,  C.  Maxwell,  R.  Lipschitz,  T.  Craig,  H.  Helmholtz,  and 
A.  B.  Basset.  Viscous  fluids  present  great  difficulties,  because  the 
equations  of  motion  have  not  the  same  degree  of  certainty  as  in  per- 
fect fluids,  on  account  of  a  deficient  theory  of  friction,  and  of  the 
dilficulty  of  connecting  oblique  pressures  on  a  small  area  with  the 
dilTerentials  of  the  velocities. 

Waves  in  liquids  have  been  a  favorite  subject  with  English  mathe- 
maticians. The  early  inquiries  of  S.  D.  Poisson  and  A.  L.  Cauchy 
were  directed  to  the  inv^estigation  of  waves  produced  by  disturbing 
causes  acting  arbitrarily  on  a  small  portion  of  the  fluid.  The  velocity 
of  the  long  wave  was  given  approximately  by  J.  Lagrange  in  1786  in 
case  of  a  channel  of  rectangular  cross-section,  by  Green  in  1839  for 
i  channel  of  triangular  section,  and  by  Philip  Kelland  (1810-1879) 


a62  a  history  of  mathematics 

of  Edinburgh  for  a  channel  of  any  uniform  section.  Sir  George  B. 
Airy,  in  his  treatise  on  Tides  and  Waves,  discarded  mere  approxima- 
tions, and  gave  the  exact  equation  on  which  the  theory  of  the  long 
wave  in  a  channel  of  uniform  rectangular  section  depends.  But  he 
gave  no  general  solutions.  J.  McCowan  of  University  College  at 
Dundee  discussed  this  topic  more  fully,  and  arrived  at  exact  and  com- 
plete solutions  for  certain  cases.  The  most  important  appHcation  of 
the  theory  of  the  long  wave  is  to  the  explanation  of  tidal  phenomena 
in  rivers  and  estuaries. 

The  mathematical  treatment  of  solitary  waves  was  first  taken  up 
by  S.  Earnshaw  in  1845,  then  by  G.  G.  Stokes;  but  the  first  sound 
approximate  theory  was  given  by  J.  Boussinesq  in  187 1,  who  obtained 
an  equation  for  their  form,  and  a  value  for  the  velocity  in  agreement 
with  experiment.  Other  methods  of  approximation  were  given  by 
Lord  Rayleigh  and  John  McCowan.  In  connection  with  deep-water 
waves,  Osborne  Reynolds  (1842-1912)  of  the  University  of  Manchester 
gave  in  1877  the  dynamical  explanation  for  the  fact  that  a  group  of 
such  waves  advances  with  only  half  the  rapidity  of  the  individual 
waves. 

The  solution  of  the  problem  of  the  general  motion  of  an  ellipsoid 
in  a  fluid  is  due  to  the  successive  labors  of  George  Green  (1833), 
R.  F.  A.  Clebsch  (1856),  and  Carl  Anton  Bjerknes  (1825-1903)  of 
Christiania  (1873).  The  free  motion  of  a  solid  in  a  liquid  has  been 
investigated  by  W.  Thomson  (Lord  Kelvin),  G.  R.  Kirchhoff,  and 
Horace  Lamb.  By  these  labors,  the  motion  of  a  single  solid  in  a  fluid 
has  come  to  be  pretty  well  understood,  but  the  case  of  two  solids  in  a 
fluid  is  not  developed  so  tully.  The  problemi  has  been  attacked  by 
W.  M.  Hicks. 

The  determination  of  the  period  of  oscillation  of  a  rotating  liquid 
spheroid  has  important  bearings  on  the  question  of  the  origin  of  the 
moon.  G.  H.  Darwin's  investigations  thereon,  viewed  in  the  light  of 
G.  F.  B.  Riemann's  and  H.  Poincare's  researches,  seem  to  disprove 
P.  S.  Laplace's  hypothesis  that  the  moon  separated  from  the  earth 
as  a  ring,  because  the  angular  velocity  was  too  great  for  stabiUty; 
G.  H.  Darwin  finds  no  instability. 

The  explanation  of  the  contracted  vein  has  been  a  point  of  much 
controversy,  but  has  been  put  in  a  much  better  light  by  the  application 
of  the  principle  of  momentum,  originated  by  W.  Froude  and  Lord 
Rayleigh.  Rayleigh  considered  also  the  reflection  of  waves,  not  at 
the  surface  of  separation  of  two  uniform  media,  where  the  transition 
is  abrupt,  but  at  the  confines  of  two  media  between  which  the  transition 
is  gradual. 

The  first  serious  study  of  the  circulation  of  winds  on  the  earth's 
surface  was  instituted  at  the  beginning  of  the  second  quarter  of  the  last 
century  by  William  C.  Redfield  (1789-1857),  an  American  meteorolo- 
gist and  railway  projector,  James  Pollard  Espy  (i 786-1860)  of  Wash- 


APPLIFT)  ^rATIIE^rATICS  463 

ington,  through  whose  stimulus  the  present  United  States  Weather 
Bureau  was  started  and  Hcinrich  Wilhelm  Dove  (1803-1879)  of  BerHn, 
followed  by  researches  by  Sir  William  Reid  (17Q1-1858)  a  British 
major-general  who  developed  his  circular  theory  of  hurricanes  while  in 
the  West  Indies,  Henry  Piddiii^lon  (1797-1858)  a  British  commander 
in  the  mercantile  marine  who  accumulated  data  for  determining  the 
course  of  storms  at  sea  and  originated  the  term  "cyclone,"  and  Ellas 
Loomis  (1S11-1889)  of  Yale  University.  But  the  deepest  insight  into 
the  wonderful  correlations  that  exist  among  the  varied  motions  of  the 
atmosphere  was  obtained  by  William  Ferrel  (1817-1891).  He  was 
born  in  Fulton  County,  Pa.,  and  brought  up  on  a  farm.  Though  in 
unfavorable  surroundings,  a  burning  thirst  for  knowledge  spurred 
the  boy  to  the  mastery  of  one  branch  after  another.  He  attended 
Marshall  College,  Pa.,  and  graduated  in  1844  from  Bethany  College. 
W^hile  teaching  school  he  became  interested  in  meteorology  and  in 
the  subject  of  tides.  In  1S56  he  wrote  an  article  on  "the  winds  and 
currents  of  the  ocean."  The  following  year  he  became  connected 
with  the  Nautical  Almanac.  A  mathematical  paper  followed  in  1858 
on  "the  motion  of  fluids  and  solids  relative  to  the  earth's  surface." 
The  subject  was  extended  afterwards  so  as  to  embrace  the  mathe- 
matical theory  of  cyclones,  tornadoes,  water-spouts,  etc.  In  1885 
appeared  his  Recent  Advances  in  Meteorology.  In  the  opinion  of 
Jidius  Hann  of  Vienna,  Ferrel  has  "contributed  more  to  the  advance 
of  the  physics  of  the  atmosphere  than  any  other  living  physicist  or 
meteorologist." 

W.  Ferrel  taught  that  the  air  flows  in  great  spirals  toward  the  poles, 
both  in  the  upper  strata  of  the  atmosphere  and  on  the  earth's  surface 
beyond  the  30th  degree  of  latitude;  while  the  return  current  blows  at 
nearly  right  angles  to  the  above  spirals,  in  the  middle  strata  as  well 
as  on  the  earth's  surface,  in  a  zone  comprised  between  the  parallels 
30°  N.  and  30°  S.  The  idea  of  three  superposed  currents  blowing  spirals 
was  first  advanced  by  James  Thomson  (1822-1892),  brother  of  Lord 
Kelvin,  but  was  published  in  very  meagre  abstract. 

W.  Ferrel's  view^s  have  given  a  strong  unpulse  to  theoretical  re- 
search in  .America,  Austria,  and  Germany.  Several  objections  raised 
against  his  argument  have  been  abandoned,  or  have  been  answered 
by  W.  M.  Da\-is  of  Harvard.  The  mathematical  analysis  of  F.  Waldo 
of  Cambridge,  ISIass.,  and  of  others,  has  further  confirmed  the  accuracy 
of  the  theory.  The  transport  of  Krakatoa  dust  and  observations  made 
on  clouds  \)o\x\t  toward  the  existence  of  an  upper  east  current  on  the 
equator,  and  Josef  M.  Pernter  (1848-1908)  of  Vienna  has  mathe- 
matically deduced  from  Ferrel's  theory  the  existence  of  such  a  current. 

Another  theory  of  the  general  circulation  of  the  atmosphere  w^as 
propounded  by  Werner  Siemens  (1816-1892)  of  Berlin,  in  which  an 
attempt  is  made  to  api)ly  thermodynamics  to  aerial  currents.  Im- 
portant /lew  points  of  view  have  been  introduced  by  H.  Helmholtz, 


464  A  HISTORY  OF  MATHEMATICS 

who  concluded  that  when  two  air  currents  blow  one  above  the  other 
in  different  directions,  a  system  of  air  waves  must  arise  in  the  same 
way  as  waves  are  formed  on  the  sea.  He  and  Anton  Oberbeck  (1846- 
1900)  of  Tubingen  showed  that  when  the  waves  on  the  sea  attain 
lengths  of  from  16  to  33  feet,  the  air  waves  must  attain  lengths  of  from 
10  to  20  miles,  and  proportional  depths.  Superposed  strata  would 
thus  mix  more  thoroughly,  and  their  energy  would  be  partly  dissipated. 
From  hydrodynamical  equations  of  rotation  H.  Helmholtz  established 
the  reason  why  the  observed  velocity  from  equatorial  regions  is  much 
less  in  a  latitude  of,  say,  20°  or  30°,  than  it  would  be  were  the  move- 
ments unchecked.  Other  important  contributors  to  the  general  theory 
of  the  circulation  of  the  atmosphere  are  Max  Moller  of  Braunschweig 
and  Luigi  de  Marchi  of  the  University  of  Pavia.  The  source  of  the 
energy  of  atmospheric  disturbances  was  sought  by  W.  Ferrel  and  Th. 
Reye  in  the  heat  given  off  during  condensation.  Max  Margules  of  the 
University  of  Vienna  showed  in  1905  that  this  heat  energy  contributes 
nothing  to  tne  kinetic  energy  of  the  winds  and  that  the  source  of 
energy  is  found  in  the  lowering  of  the  centre  of  gravity  of  an  air  column 
when  the  colder  air  assumes  the  lower  levels,  whereby  the  potential 
energy  is  diminished  and  the  kinetic  energy  increased.^  Asymmetric 
cyclones  have  been  studied  especially  by  Luigi  de  Marchi  of  Pavia. 
Anticyclones  have  received  attention  from  Henry  H.  Clayton  of  the 
Blue  Hill  Observatory,  near  Boston,  from  Julius  Hann  of  Vienna,  F. 
H.  Bigelow  of  Washington,  and  Max  Margules  of  Vienna. 

Sound.     Elasticity 

About  i860  acoustics  began  to  be  studied  with  renewed  zeal.  The 
mathematical  theory  of  pipes  and  vibrating  strings  had  been  elabo- 
rated in  the  eighteenth  century  by  Daniel  Bernoulli,  D'Alembert, 
L.  Euler,  and  J.  Lagrange.  In  the  first  part  of  the  present  century 
P.  S.  Laplace  corrected  Newton's  theory  on  the  velocity  of  sound  in 
gases;  S.  D.  Poisson  gave  a  mathematical  discussion  of  torsional 
vibrations;  S.  D.  Poisson,  Sophie  Germain,  and  Charles  Wheatstone 
studied  Chladni's  figures;  Thomas  Young  and  the  brothers  Weber 
developed  the  wave-theory  of  sound.  Sir  J.  F.  W.  Herschel  (1792- 
1871)  wrote  on  the  mathematical  theory  of  sound  for  the  Encyclo- 
paedia Metropolitana,  1S45.  Epoch-making  were  H.  Helmholtz's 
experimental  and  mathematical  researches.  In  his  hands  and  Ray- 
leigh's,  Fourier's  series  received  due  attention.  H.  Helmholtz  gave 
the  mathematical  theory  of  beats,  difference  tones,  and  summation 
tones.  Lord  Rayleigh  (John  William  Strutt)  of  Cambridge  (bom 
1842)  made  extensive  mathematical  researches  in  acoustics  as  a  part 
of  the  theory  of  vibration  in  general.  Particular  mention  may  be 
made  of  his  discussion  of  the  disturbance  produced  by  a  spherical 

'  Encyklopiidie  der  Math.  Wissenschaflen ,  Bd.  VI,  i,  8,  1912,  p.  216. 


APPLIED  MATHEMATICS  465 

(Obstacle  on  the  waves  of  sound,  and  of  phenomena,  such  as  sensitive 
flames,  connected  with  the  instability  of  jets  of  fluid.  In  1877  and  187S 
he  published  in  two  volumes  a  treatise  on  The  Theory  of  Sound.  Other 
mathematical  researches  on  this  subject  have  been  made  in  England 
by  William  Ushburn  Donkin  (1814-1869)  of  Oxford  and  G.  G.  Stokes. 
An  interesting  point  in  the  behavior  of  a  Fourier's  series  was  brought 
out  in  1898  by  J.  W.  Gibbs  of  Yale.  A.  A.  Michelson  and  S.  W.  Strat- 
ton  at  the  University  of  Chicago  had  shown  experimentally  by  their 
harmonic  analyses  that  the  summation  of  160  terms  of  the  series 
2(— i)i"+H^''«rix)/«  revealed  certain  unexpected  small  towers  in  the 
curve  for  the  sum,  as  n  increased.  J.  W.  Gibbs  showed  (Nature,  Vol. 
59,  p.  606)  by  the  study  of  the  order  of  variation  of  n  and  .r  that  these 
phenomena  were  not  due  to  imperfections  in  the  machine,  but  were 
true  mathematical  phenomena.  They  are  called  the  "  Gibbs'  phenom- 
enon," and  have  received  further  attention  from  Maxime  Bocher, 
T.  H.  Gronwall,  H.  Weyl,  and  H.  S.  Carslaw. 

The  theory  of  elasticity  ^  belongs  to  this  century.  Before  1800  no 
attempt  had'  been  made  to  form  general  equations  for  the  motion  or 
equilibrium  of  an  elastic  solid.  Particular  problems  had  been  solved 
by  special  h^-potheses.  Thus,  James  Bernoulli  considered  elastic 
laminae;  Daniel  Bernoulli  and  L.  Euler  investigated  vibrating  rods; 
J.  Lagrange  and  L.  Euler,  the  equilibrium  of  springs  and  columns. 
The  earliest  investigations  of  this  century,  by  Thomas  Young 
("Young's  modulus  of  elasticity")  in  England,  J.  Binet  in  France, 
and  G.  A.  A.  Plana  in  Italy,  were  chiefly  occupied  in  extending  and 
correcting  the  earlier  labors.  Between  1820  and  1840  the  broad  out- 
line of  the  modem  theory  of  elasticity  was  established.  This  was  ac- 
complished almost  exclusively  by  French  writers, — Louis-Marie- 
Henri  Navier  (1785-1S36),  S.  D.  Poisson,  A.  L.  Cauchy,  Mademoiselle 
Sophie  Germain  (1776-1831),  Felix  Savart  (1791-1841).  Says  H. 
Burkhardt:  "There  are  two  views  respecting  the  beginnings  of  the 
theory  of  elasticity  of  solids,  of  which  no  dimension  can  be  neglected : 
According  to  one  \\q\w  the  deciding  impulse  came  from  Fresnel's 
undulatory  theory  of  light,  according  to  the  other,  everything  goes 
back  to  the  technical  theory  of  rigidity  (Festigkeitstheorie),  the  rep- 
resentative of  which  was  at  that  time  Navier.  As  always  in  such 
cases,  the  truth  lies  in  the  middle:  Cauchy  to  whom  we  owe  primarily 
the  fixing  of  the  fundamental  concepts,  as  strain  and  stress,  learned 
from  Fresnel  as  well  as  from  Navier." 

Simeon  Denis  Poisson  -  (i 781-1840)  was  bom  at  Pithiviers.  The 
boy  was  put  out  to  a  nurse,  and  he  used  to  tell  that  when  his  father 
(a  common  soldier)  came  to  see  him  one  day,  the  nurse  had  gone  out 

'  T.  Todhunter,  History  of  the  Theory  of  Elaslicily,  edited  by  Karl  Pearson,  Cam- 
bridge, 1886.  '  ,   .        . 

2  Ch.  Hermite,   "Disrours  prononc^  devant  le  president  de  la   Republiqiu'. 
Bulletin  des  sciences  malhematiques  XIV  Janvier,  1890. 


4^.6  A  HISTORY  OF  MATHEMATICS 

and  left  him  suspended  by  a  thin  cord  to  a  nail  in  the  wall  in  order  to 
protect  him  from  perishing  mider  the  teeth  of  the  carnivorous  and  un- 
clean animals  that  roamed  on  the  floor.  Poisson  used  to  add  that  his 
gymnastic  elTorts  when  thus  suspended  caused  him  to  swing  back  and 
forth,  and  thus  to  gain  an  early  familiarity  with  the  pendulum,  the 
study  of  wliich  occupied  him  much  in  his  maturer  life.  His  father 
destined  him  for  the  medical  profession,  but  so  repugnant  was  this 
to  him  that  he  was  permitted  to  enter  the  Polytechnic  School  at  the 
age  of  seventeen.  His  talents  excited  the  interest  of  J.  Lagrange  and 
P.  S.  Laplace.  At  eighteen  he  ■wrote  a  memoir  on  finite  differences 
which  was  printed  on  the  recommendation  of  A.  M.  Legendre.  He 
soon  became  a  lecturer  at  the  school,  and  continued  through  Hfe  to 
hold  various  government  scientific  posts  and  professorships.  He  pre- 
pared some  400  publications,  mainly  on  applied  mathematics.  His 
Traite  de  Mecamque,  2  vols.,  181 1  and  1833,  was  long  a  standard  work. 
He  wrote  on  the  mathematical  theory  of  heat,  capillary  action,  proba- 
bility of  judgment,  the  mathematical  theory  of  electricity  and  mag- 
netism, physical  astronomy,  the  attraction  of  ellipsoids,  definite  in' 
tegrals,  series,  and  the  theory  of  elasticity.  He  was  considered  one 
of  the  leading  analysts  of  his  time.  The  story  is  told  that  in  1802  a 
young  man,  about  to  enter  the  army,  asked  Poisson  to  take  $100  in 
safe-keeping.  "All  right,"  said  Poisson,"  set  it  down  there  and  let 
me  work;  I  have  much  to  do."  The  recruit  placed  the  money-bag  on 
a  shelf  and  Poisson  placed  a  copy  of  Horace  over  the  bag,  to  hide  it. 
Twenty  years  later  the  soldier  returned  and  asked  fcr  his  money, 
but  Poisson  remembered  nothing  and  asked  angrily:  "You  claim 
to  have  put  the  money  in  my  hands?"  "No,"  repUed  the  soldier, 
"I  put  in  on  this  shelf  and  you  placed  this  book  over  it."  The  soldier 
removed  the  dusty  copy  of  Horace  and  found  the  $100  where  they  had 
been  placed  twenty  years  before. 

His  work  on  elasticity  is  hardly  excelled  by  that  of  A.  L.  Cauchy, 
and  second  only  to  that  of  B.  de  Saint-Venant.  There  is  hardly  a 
problem  in  elasticity  to  which  he  has  not  contributed,  while  many  of 
his  inquiries  were  new.  The  equilibrium  and  motion  of  a  circular  plate 
was  first  successfully  treated  by  him.  Instead  of  the  definite  integrals 
of  earlier  writers,  he  used  preferably  finite  summations.  Poisson's 
contour  conditions  for  elastic  plates  were  objected  to  by  Gustav 
Kirchhoff  of  Berlin,  who  established  new  conditions.  But  Thomson 
(Lord  Kelvin)  and  P.  G.  Tait  in  their  Treatise  on  Natural  Philosophy 
have  explained  the  discrepancy  between  Poisson's  and  Kirchhofl's 
boundary  conditions,  and  established  a  reconciliation  between  them. 

Important  contributions  to  the  theory  of  elasticity  were  made  by 
A.  L.  Cauchy.  To  him  we  owe  the  origin  of  the  theory  of  stress,  and 
the  transition  from  the  consideration  of  the  force  upon  a  molecule 
exerted  by  its  neighbors  to  the  consideration  of  the  stress  upon  a 
small  plane  at  a  point.    He  anticipated  G.  Green  and  G.  G.  Stokes 


APPTJEl^  MATHEMATICS  467 

in  giving  the  equations  of  isotropic  elasticity  with  two  constants. 
The  theory  of  elasticity  was  presented  by  Gabrio  Piola  of  Italy  ac- 
cording to  the  principles  of  J.  Lagrange's  M echaniqiie  Analytique,  but 
the  superiority  of  this  method  over  that  of  Poisson  and  Cauchy  is  far 
from  evident.  The  influence  of  tem]icrature  on  stress  was  first  in- 
vestigated experimentally  by  Wilhelm  Weber  of  Gottingen,  and 
afterwards  mathematically  by  J.  M.  C.  Duhamel,  who,  assuming 
Poisson's  theory  of  elasticity,  examined  the  alterations  of  form  which 
the  formula^  undergo  when  we  allow  for  changes  of  temperature.  W. 
Weber  was  also  the  first  to  ex-periment  on  elastic  after-strain.  Other 
important  experiments  were  made  by  different  scientists,  which  dis- 
closed a  -vender  range  of  phenomena,  and  demanded  a  more  compre- 
hensive theory.  Set  was  investigated  by  Franz  Joseph  von  Gerstner 
(1756-1832),  of  Prague  and  Eaton  Hodgkinson  of  University  College, 
London,  while  the  latter  physicist  in  England  and  Louis  Joseph  Vicat 
(i  786-1861)  in  Erance  experimented  extensively  on  absolute  strength. 
L.  J.  Vicat  boldly  attacked  the  mathematical  theories  of  flexure  be- 
cause they  failed  to  consider  shear  and  the  time-element.  As  a  result, 
a  truer  theory  of  flexure  was  soon  propounded  by  B.  de  Saint- Venant. 
J.  V.  Poncelet  advanced  the  theories  of  resihence  and  cohesion. 

Gabriel  Lame  (1795-1870)  was  bom  at  Tours,  and  graduated  at 
the  Polytechnic  School.  He  was  called  to  Russia  with  B.  P.  E.  Clap- 
eyron  and  others  to  superintend  the  construction  of  bridges  and  roads. 
On  his  return,  in  1832,  he  was  elected  professor  of  physics  at  the  Poly- 
technic School.  Subsequently  he  held  various  engineering  posts  and 
professorships  in  Paris.  As  engineer  he  took  an  active  part  in  the  con- 
struction of  the  first  railroads  in  Erance.  Lame  devoted  his  fine  mathe- 
matical talents  mainly  to  mathematical  physics.  In  four  works: 
Lemons  sur  les  fondions  inverses  des  transcendantes  et  les  surfaces  isother- 
mes;  Sur  les  coordonnees  curvilignes  et  leurs  diverses  applications;  Sur 
la  theorie  analytique  de  la  chaleur;  Sur  la  theorie  mathematique  de  Velas- 
ticite  des  corps  solides  (1S52),  and  in  various  memoirs  he  displays  fine 
analytical  powers;  but  a  certain  want  of  physical  touch  sometimes  re- 
duces the  value  of  his  contributions  to  elasticity  and  other  physical 
subjects.  In  considering  the  temperature  in  the  interior  of  an  ellip- 
soid under  certain  conditions,  he  employed  functions  analogous  to  La- 
place's functions,  and  kno\\Ti  by  the  name  of  "Lame's  functions." 
A  problem  in  elasticity  called  by  Lame's  name,  viz.,  to  investigate 
the  conditions  for  equilibrium  of  a  spherical  elastic  envelope  subject 
to  a  given  distribution  of  load  on  the  bounding  spherical  surfaces,  and 
the  determination  of  the  resulting  shifts  is  the  only  completely  general 
problem  on  elasticity  wliich  can  be  said  to  be  completely  solved.  He 
deserves  much  credit  for  his  derivation  and  transformation  of  the 
general  elastic  equations,  and  for  his  api)lication  of  them  to  double 
refraction.  Rectangular  and  triangular  membranes  were  shown  by 
him  to  be  connected  with  questions  in  the  theory  of  numbers.     H< 


468  A  HISTORY  OF  MATHEMATICS 

Burkhardt  ^  is  of  the  opinion  that  the  importance  of  the  classic 
period  of  French  mathematical  physics,  about  1810-1835,  is  often 
undervalued,  but  that  the  direction  it  took  finally  under  the  leader- 
ship of  Lame  was  unfortunate.  "By  his  (Lame's)  taste  for  algebraic 
elegance  he  was  misled  to  prefer  problems  which  are  of  interest  in  pure 
rather  than  applied  mathematics;  he  went  so  far  as  to  require  of  tech- 
nical men  the  study  of  number  theory,  because  the  determination  of 
the  simple  tones  of  a  rectangular  plate  with  commensurable  sides 
calls  for  the  solution  of  an  indeterminate  quadratic  equation." 

Continuing  our  outline  of  the  history  of  elasticity,  we  observe  that 
the  field  of  photo-elasticity  was  entered  upon  by  G.  Lame,  F.  E.  Neu- 
mann, and  Clerk  Maxwxll.  G.  G.  Stokes,  W.  Wertheim,  R.  Clausius, 
and  J.  H.  Jellett,  threw  new  light  upon  the  subject  of  " rari-constancy  " 
and  "multi-constancy,"  which  has  long  divided  elasticians  into  two  op- 
posing factions.  The  uni-constant  isotropy  of  L.  M.  H.  Navier  and 
S.  D.  Poisson  had  been  questioned  by  A.  L.  Cauchy,  and  was  severely 
criticised  by  G.  Green  and  G.  G.  Stokes. 

Barre  de  Saint-Venant  (1797-18S6),  ingenieur  des  ponts  et  chaus- 
sees,  made  it  his  life-work  to  render  the  theory  of  elasticity  of  prac- 
tical value.  The  charge  brought  by  practical  engineers,  like  Vicat, 
against  the  theorists  led  Saint-Venant  to  place  the  theory  in  its  true 
place  as  a  guide  to  the  practical  man.  Numerous  errors  committed 
by  his  predecessors  were  removed.  He  corrected  the  theory  of  flexure 
by  the  consideration  of  slide,  the  theory  of  elastic  rods  of  double 
curvature  by  the  introduction  of  the  tliird  moment,  and  the  theory 
of  torsion  by  the  discovery  of  the  distortion  of  the  primitively  plane 
section.  His  results  on  torsion  abound  in  beautiful  graphic  illustra- 
tions. In  case  of  a  rod,  upon  the  side  surfaces  of  which  no  forces  act, 
he  showed  that  the  problems  of  flexure  and  torsion  can  be  solved, 
if  the  end-forces  are  distributed  over  the  end-surfaces  by  a  definite 
law.  R.  F.  A.  Clebsch,  in  his  Lehrbuch  der  Elasticitat,  1862,  showed 
that  this  problem  is  reversible  to  the  case  of  side-forces  without  end- 
forces.  Clebsch  -  extended  the  research  to  very  thin  rods  and  to  very 
thin  plates.  B.  de  Saint-Venant  considered  problems  arising  in  the 
scientific  design  of  built-up  artillery,  and  his  solution  of  them  differs 
considerably  from  G.  Lame's  solution,  which  was  popularized  by  W.  J. 
M.  Rankine,  and  much  used  by  gun-designers.  In  Saint- Venant's 
translation  into  French  of  Clebsch's  Elasticitat,  he  develops  extensively 
a  double-suffix  notation  for  strain  and  stresses.  Though  often  advan- 
tageous, this  notation  is  cumbrous,  and  has  not  been  generally  adopted. 
Karl  Pearson,  Galton  professor  of  eugenics  at  the  University  of  London, 
in  his  early  mathematical  studies,  examined  the  permissible  limits  of 
the  application  of  the  ordinary  theory  of  flexure  of  a  beam. 

^  Jahrcsb.  d.  d.  Math.  Vernnigiing,  Vol.  12,  1903,  p.  564. 

2  Alfred  Clebsch,  Versiuh  einer  Darlegiing  iin/i  Wurdigiing  seiner  imssenschaftlichen 
Leislungen  von  einigen  seiner  Freunde,  Leipzig,  1873. 


APPLIED  MAIUKMATirs  469 

The  mathematical  theory  of  elasticity  is  still  in  an  unsettled  con- 
dition. Not  only  are  scientists  still  divided  into  two  schools  of  "rari- 
constancy"  and  "multi-constancy,"  but  difference  of  opinion  exists 
on  other  vital  questions.  Among  the  numerous  modern  writers  on 
elasticity  may  be  mentioned  Emile  Mathieu  (1835-1890),  professor 
at  Besan^on,  Maurice  Le\y  (1838-1910)  of  the  College  de  France  in 
Paris,  Charles  Chree,  superintendent  of  the  Kew  Observatory^,  A.  B. 
Basset,  Lord  Kelvin  of  Glasgow,  J.  Boussinesq  of  Paris,  and  others. 
Lord  Kelvin  applied  the  laws  of  elasticity  of  solids  to  the  investigation 
of  the  earth's  elasticity,  which  is  an  important  element  in  the  theory 
of  ocean-tides.  If  the  earth  is  a  solid,  then  its  elasticity  co-operates 
with  gravity  in  opposing  deformation  due  to  the  attraction  of  the  sun 
and  moon.  P.  S.  Laplace  had  shown  how  the  earth  would  behave  if  it 
resisted  deformation  only  by  gravity.  G.  Lame  had  investigated  how 
a  solid  sphere  would  change  if  its  elasticity  only  came  into  play. 
Lord  Kelvin  combined  the  two  results,  and  compared  them  with  the 
actual  deformation.  Kelvin,  and  afterw^ards  G.  H.  Darwin,  computed 
that  the  resistance  of  the  earth  to  tidal  deformation  is  nearly  as  great 
as  though  it  were  of  steel.  This  conclusion  was  confirmed  more  re- 
cently by  Simon  Newcomb,  from  the  study  of  the  observed  periodic 
changes  in  latitude  and  by  others.  Fcr  an  ideally  rigid  earth  the 
period  would  be  360  days,  but  if  as  rigid  as  steel,  it  would  be  441,  the 
observed  period  being  430  days. 

Among  the  older  text-books  on  elasticity  may  be  mentioned  the 
works  of  G.  Lame,  R.  F.  A.  Clcbsch,  A.  Winckler,  A.  Beer,  E.  L. 
Mathieu,  W.  J.  Ibbetson,  and  F.  Neumann,  edited  by  O.  E.  Meyer. 

In  recent  years  the  modem  analytical  developments,  particularly 
along  the  line  of  integral  ecjuations,  have  been  brought  to  bear  on 
theories  of  elasticity  and  potential.  The  solution  of  the  static  prob- 
lem of  the  theory  of  elasticity  of  a  homogeneous  isotropic  body  under 
certain  given  surface  conditions  has  been  taken  up  particularly  by 
E.  I.  Fredholm  of  Stockholm,  G.  Lauricclla  of  the  University  of 
Catania,  R.  Marcolongo  of  Naples  and  Hermann  Weyl  of  Zurich, 
and  by  a  somewhat  different  mode  of  procedure,  by  A.  Kom  of  Berlin 
and  T.  Boggio  of  Turin. ^ 

Closely  connected  with  researches  on  attraction  and  elasticitv  is 
the  development  of  spherical  harmonics.  After  the  initial  paper  of 
A.  M.  Legendre  on  zonal  harmonics  applied  by  him  to  the  stud}'  of 
the  attraction  of  solids  of  revolu' 'on,  and  after  the  remarkable  memoir 
of  1782  by  P.  S.  Laplace  who  used  spherical  harmonics  in  finding  the 
potential  of  a  solid  nearly  spherical,  the  first  ad\ance  was  made  by 
Olinde  Rodrigues  (1794-1851),  a  French  economist  and  reformer, 
who  in  1 8 16  gave  a  formula  for  P«  which  later  was  derived  independ- 
ently by  J.  Ivor>'  and  C.  G.  J.  Jacobi.  The  name  "Kugelfunktion" 
is  due  to  K.  F.  Gauss.  Important  contributions  were  made  in  G:r- 
1  See  Rendiconli  del  Circolo  Math,  di  Palmno,  Vol.  39,  1915,  p.  i. 


470  A  HISTORY  OF  MATHEMATICS 

many  by  C.  G.  J.  Jacobi,  L.  Dirichlct,  Franz  Ernst  Neumann  (1798- 
1S95)  who  was  professor  of  physics  and  mineralogy  in  Konigsberg, 
his  son  Carl  Neumann  (1832-  ),  Elwin  Bruno  Christoffel  (1829- 
1900)  of  the  University  of  Strassburg,  R.  Dedekind,  Gustav  Bauer 
(1820-1906)  of  Munich,  Gustav  Mehler  (1835-1895)  of  Elbingin  West 
Prussia,  and  Karl  Baer  (1851-  )  of  Kiel.  Especially  active  was 
Eduard  Heine  (1S21-1SS1)  of  the  University  of  Halle,  the  author  of 
the  Handbuch  der  Kugclfiuiktionen,  1861,  2.  Ed.  1S7S-1881.  The  chief 
representative  in  the  cultivation  of  this  subject,  in  Switzerland,  was 
L.  Schlafli  of  the  University  of  Bern;  in  Belgium,  was  Eugene  Catalan 
of  the  University  of  Liege;  in  Italy,  was  E.  Beltrami;  in  the  United 
States,  was  W.  E.  Byerly  of  Harvard  University.  In  France  there 
were  S.  D.  Poisson,  G.  Lame,  T.  J.  Stieltjes,  J.  G.  Darboux,  Ch. 
Hermite,  Paul  Mathicu,  Hermann  Laurent  (1841-1908),  Professor  at 
the  Polytechnic  School  in  Paris  whose  researches  gave  rise  to  contests 
of  priority  with  German  writers.  In  Great  Britain  spherical  har- 
monics received  the  attention  of  Thomson,  (Lord  Kelvin)  and  P.  G. 
Tait  in  their  Natural  Philosophy  of  1867,  and  of  Sir  William  D.  Niven 
of  Manchester,  Norman  Ferrers  (1829-1903)  of  Cambridge,  E.  W. 
Hobson  of  Cambridge,  A.  E.  H.  Love  of  Oxford,  and  others. 

Light,  Electricity,  Heat,  Potential 

G.  F.  B.  Riemann's  opinion  that  a  science  of  physics  only  exists  since 
the  invention  of  differential  equations  finds  corroboration  even  in  this 
brief  and  fragmentary  outline  of  the  progress  of  mathematical  physics. 
The  undulatory  theory  of  light,  first  advanced  by  C.  Huygens,  owes 
much  to  the  power  of  mathematics:  by  mathematical  analysis  its 
assumptions  were  worked  out  to  their  last  consequences.  Thomas 
Young  ^  (1773-1829)  was  the  first  to  explain  the  principle  of  inter- 
ference, both  of  light  and  sound,  and  the  first  to  bring  forward  the 
idea  of  transverse  vibrations  in  light  waves.  T.  Young's  explanations, 
not  being  verified  by  him  by  extensive  numerical  calculations,  at- 
tracted little  notice,  and  it  was  not  until  Augustin  Fresnel  (1788- 
1827)  applied  mathematical  analysis  to  a  much  greater  extent  than 
Young  had  done,  that  the  undulatory  theory  began  to  carry  convic- 
tion. Some  of  Fresnel's  mathematical  assumptions  were  not  satis- 
factory; hence  P.  S.  Laplace,  S.  D,  Poisson,  and  others  belonging  to 
the  strictly  mathematical  school,  at  first  disdained  to  consider  the 
theory.  By  their  opposition  Fresnel  was  spurred  to  greater  exertion. 
D.  F.  J,  Arago  was  the  first  great  convert  made  by  Fresnel.  When 
polarization  and  double  refraction  were  explained  by  T.  Young  and 
A.  Fresnel,  then  P.  S.  Laplace  was  at  last  won  over.  S.  D.  Poisson 
drew  from  Fresnel's  formulae  the  seemingly  paradoxical  deduction 

1  Arthur  Schuster,  "Tlie  rnlliience  of  Mathcnuitics  on  the  Progress  of  Physics," 
Nature,  Vol.  25,  1882,  p.  ^^98. 


APPLIED  MATHEMATICS  471 

that  a  small  circular  disc,  illuminated  by  a  luminous  point,  must  cast 
a  shadow  with  a  bright  spot  in  the  centre.  But  this  was  found  to  be 
in  accordance  with  fact.  The  theory  was  taken  up  by  another  great 
mathematician,  W.  R.  Hamilton,  who  from  his  formulae  predicted 
conical  refraction,  verified  experimentally  by  Humphrey  Lloyd.  These 
predictions  do  not  prove,  however,  that  Fresnel's  formulae  are  correct, 
for  these  prophecies  might  have  been  made  by  other  forms  of  the 
wave-theory.  The  theory  was  placed  on  a  sounder  dynamical  basis 
by  the  writings  of  A.  L.  Cauchy,  J.  B.  Biot,  G.  Green,  C.  Neumann, 
G.  R.  Kirchhoff,  J.  MacCullagh,  G.  G.  Stokes,  B.  de  Saint-Venant, 
Emile  Sarrau  (1S37-1904)  of  the  Polytechnic  School  in  Paris,  Ludwig 
Lorenz  (1S29-1S91)  of  Copenhagen,  and  Sir  William  Thomson  (Lord 
Kelvin).  In  the  wave-theory,  as  taught  by  G.  Green  and  others,  the 
luminiferous  ether  was  an  incompressible  elastic  solid,  for  the  reason 
that  fluids  could  not  propagate  transverse  vibrations.  But,  according 
to  G.  Green,  such  an  elastic  solid  would  transmit  a  longitudinal  dis- 
turbance with  infinite  velocity.  G.  G.  Stokes  remarked,  however,  that 
the  ether  might  act  like  a  fluid  in  case  of  finite  disturbances,  and  like 
an  elastic  solid  in  case  of  the  infinitesimal  disturbances  in  light  prop- 
agation. A.  Fresnel  postulated  the  density  of  ether  to  be  different  in 
different  media,  but  the  elasticity  the  same,  while  C.  Neumann  and 
J.  MacCullagh  assumed  the  density  uniform  and  the  elasticity  different 
in  all  substances.  On  the  latter  assumption  the  direction  of  vibration 
lies  in  the  plane  of  polarization,  and  not  perpendicular  to  it,  as  in  the 
theory  of  A.  Fresnel. 

While  the  above  writers  endeavored  to  explain  all  optical  properties 
of  a  medium  on  the  supposition  that  they  arise  entirely  from  difference 
in  rigidity  or  density  of  the  ether  in  the  medium,  there  is  another 
school  advancing  theories  in  which  the  mutual  action  between  the 
molecules  of  the  body  and  the  ether  is  considered  the  main  cause  of 
refraction  and  dispersion.^  The  chief  workers  in  this  field  were  J. 
Boussinesq,  W.  Sellmeyer,  H.  Helmholtz,  E.  Lommel,  E.  Ketteler, 
W.  Voigt,  and  Sir  William  Thomson  (Lord  Kelvin)  in  his  lectures 
delivered  at  the  Johns  Hopkins  University  in  1S84.  Neither  this  nor 
the  first-named  school  succeeded  in  explaining  all  the  phenomena. 
A  third  school  was  founded  by  C.  Maxwell.  He  proposed  the  electro- 
magnetic theory,  which  has  received  extensive  development  recently. 
It  will  be  mentioned  again  later.  According  to  Maxwell's  theory,  the 
direction  of  vibration  does  not  lie  exclusively  in  the  plane  of  polariza- 
tion, nor  in  a  plane  perpendicular  to  it,  but  something  occurs  in  both 
planes — a  magnetic  vibration  in  one,  and  an  electric  in  the  other. 
G.  F.  Fitzgerald  and  F.  T.  Trouton  in  Dublin  verified  this  conclusion 
of  C.  Maxwell  by  experiments  on  electro-magnetic  waves. 

Of  recent  mathematical  and  experimental  contributions  to  optics, 

'  R.  T.  C'azebrook,  "Report  on  Optical  Theories,"  Report  British  Ass'n  for  1885, 
p.  2x3. 


472  A  HISTORY  OF  MATHEMATICS 

mention  must  be  made  of  Henry  Augustus  Rowland  (1848-1901),  who 
was  professor  of  physics  at  the  Johns  Hopkins  University  and  his 
theory  of  concave  gratings,  and  of  A.  A.  Michelson's  work  on  interfer- 
ence, and  his  application  of  interference  methods  to  astronomical 
measurements. 

A  function  of  fundamental  importance  in  the  mathematical  theories 
of  electricity  and  magnetism  is  the  "potential."  It  was  first  used  by 
J.  Lagrange  in  the  determination  of  gravitational  attractions  in  1773. 
Soon  after,  P.  S.  Laplace  gave  the  celebrated  differential  equation, 

which  was  extended  by  S.  D.  Poisson  by  writing— 4 tt^  in  place  of 
zero  in  the  right-hand  member  of  the  equation,  so  that  it  appUes  not 
only  to  a  point  external  to  the  attracting  mass,  but  to  any  point  what- 
ever. The  first  to  apply  the  potential  function  to  other  than  gravita- 
tion problems  was  George  Green  (1793-1841).  He  introduced  it  into 
the  mathematical  theory  of  electricity  and  magnetism.  Green  was  a 
self-educated  man  who  started  out  as  a  baker,  and  at  his  death  was 
fellow  of  Caius  College,  Cambridge.  In  1828  he  published  by  private 
subscription  at  Nottingham  a  paper  entitled  Essay  on  the  application 
of  mathematical  analysis  to  the  theory  oj  electricity  and  magnetism. 
About  100  copies  were  printed.  It  escaped  the  notice  even  of  English 
mathematicians  until  1S46,  when  William  Thomson  (Lord  Kelvin)  had 
it  reprinted  in  Crelle's  Journal,  vols.  xliv.  and  xlv.  It  contained  what  is 
now  known  as  "Green's  theorem"  for  the  treatment  of  potential. 
Meanwhile  all  of  Green's  general  theorems  had  been  rediscovered  by 
William  Thomson  (Lord  Kelvin),  M.  Chasles,  J.  C.  F.  Sturm,  and 
K.  F.  Gauss.  The  term  potential  function  is  due  to  G.  Green.  W.  R. 
Hamilton  used  the  word  force-function,  while  K.  F.  Gauss,  who  about 
1840  secured  the  general  adoption  of  the  function,  called  it  simply 
potential.  G.  Green  wrote  papers  on  the  equilibrium  of  fluids,  the 
attraction  of  ellipsoids,  on  the  reflection  and  refraction  of  sound  and 
light.  His  researches  bore  on  questions  previously  considered  by 
S.  D.  Poisson.  K.  F.  Gauss  proved  what  C.  Neumann  has  called 
"Gauss'  theorem  of  mean  value"  and  then  considered  the  question  of 
maxima  and  minima  of  the  potential.^ 

Large  contributions  to  electricity  and  magnetism  have  been  made 
by  William  Thomson  later  Sir  William  Thomson  and  Lord  Kelvin 
(1824-1907).  He  was  born  at  Belfast,  Ireland,  but  was  of  Scotch  de- 
scent. He  and  his  brother  James  studied  in  Glasgow.  From  there  he 
entered  Cambridge,  and  was  graduated  as  Second  Wrangler  in  1845. 
William  Thomson,  J.  J.  Sylvester,  C.  Maxwell,  W.  K.  ClilTord,  and  J.  j. 
Thomson  are  a  group  of  great  men  who  were  Second  Wranglers  at  Cam- 
bridge.   At  the  age  of  twenty-two  W.  Thomson  was  elected  professor 

'  For  details  see  Max  Bacharach,  Geschichle  der  Potenliallheorie,  Gottingen,  1883. 


AITLIKH  MATH  KM  All  CS 


473 


of  natural  philosophy  in  the  University  of  Glasgow,  a  position  which 
he  held  till  his  death.  For  his  brilHant  mathematical  and  physical 
achievements  he  was  knighted,  and  in  iSq2  was  made  Lord  Kelvin. 
He  was  greatly  influenced  by  the  mathematical  physics  of  J.  Fourier 
and  other  French  mathematicians.  It  was  Fourier's  mathematics  on 
the  flow  of  heat  through  solids  which  led  him  to  the  mastery  of  the 
diffusion  of  an  electric  current  through  a  wire  and  of  the  difhculties 
encountered  in  signalling  through  the  Atlantic  telegraph.  In  1845 
W.  Thomson  visited  Paris.  P.  S.  Laplace,  A.  M.  Legendre,  J.  Fourier, 
Sadi  Carnot,  S.  D.  Poisson,  and  A.  Fresnel  were  no  longer  living.  W. 
Thomson  met  J.  Liouville  to  whom  he  gave  the  now  famous  memoir 
of  G.  Green  of  the  year  1828.  He  met  M.  Chasles,  J.  B.  Biot,  H.  V. 
Regnault,  J.  C.  F.  Sturm,  A.  L.  Cauchy,  and  J.  B.  L.  Foucault.  A.  L 
Cauchy  tried  to  convert  him  to  Roman  Catholicism.  One  evening. 
J.  C.  F.  Sturm  called  upon  him  in  high  excitement.  "Vous  avez  le 
memoire  de  Green,"  he  exclaimed.  The  Essay  was  produced;  Sturm 
eagerly  scanned  its  contents.  "Ah!  voila  mon  alTaire,"  he  cried, 
jumping  from  his  seat  as  he  caught  sight  of  the  formula  in  which  G. 
Green  had  anticipated  his  theorem  of  the  equivalent  distribution. 
Kelvin's  researches  on  the  theory  of  potential  are  epoch-making. 
What  is  called  "Dirichlet's  principle"  was  discovered  by  him  in  1848, 
somewhat  earlier  than  by  P.  G.  L.  Dirichlet.  Jointly  with  P.  G.  Tait 
he  prepared  the  celebrated  Treatise  of  Natural  Philosophy,  1867. 
As  a  mathematician  he  belonged  most  decidedly  to  the  intuitional 
school.  Purists  in  mathematics  often  carped  at  Kelvin's  "instinctive " 
mathematics.  "Do  not  imagine,"  he  once  said,  "that  mathematics 
is  hard  and  crabbed,  and  repulsive  to  common  sense.  It  is  merely  the 
etherealization  of  common  sense."  Yet  even  in  mathematics  he  had 
his  dislikes.  When  in  1845  he  met  W.  R.  Hamilton  at  a  British  Asso- 
ciation meeting,  who  then  read  his  first  paper  on  Quaternions,  one 
might  have  thought  that  W.  Thomson  would  welcome  the  new 
analysis:  but  it  was  not  so.  He  did  not  use  it.  On  the  merits  of  quater- 
nions he  had  a  thirty-eight  years'  war  with  P.  G.  Tait.^  We  owe  to 
W.  Thomson  new  synthetical  methods  of  great  elegance,  viz.,  the 
theory  of  electric  images  and  the  method  of  electric  inversion  founded 
thereon.  By  them  he  determined  the  distribution  of  electricity  on  a 
bowl,  a  problem  previously  considered  insolvable.  The  distribution 
of  static  electricity  on  conductors  had  been  studied  before  this  mainly 
by  S.  D.  Poisson  and  G.  A,  A.  Plana.  In  1845  F.  E.  Neumann  of 
Konigsberg  developed  from  the  experimental  laws  of  Lenz  the  mathe- 
matical theory  of  magneto-electric  induction.  In  1855  W.  Thomson 
predicted  by  mathematical  analysis  that  the  discharge  of  a  Leyden  jar 
through  a  linear  conductor  would  in  certain  cases  consist  of  a  series  of 
decaying  oscillations.  This  was  first  established  experimentally  by 
Joseph  Henry  of  Washington.  William  Thomson  worked  out  the 
'  S.  P.  Thompson,  Life  of  William  Thomson,  London,  1910,  pp.  452,  1 136-1139. 


474  A  HISTORY  OF  MATHEMATICS 

electro-static  induction  in  submarine  cables.  The  subject  of  tht 
screening  effect  against  induction,  due  to  sheets  of  different  metals, 
was  worked  out  mathematically  by  Horace  Lamb  and  also  by  Charles 
Niven.  W.  Weber's  chief  researches  were  on  electro-dynamics.  H. 
Helmholtz  in  1S51  gave  the  mathematical  theory  of  the  course  of  in- 
duced currents  in  various  cases.  Gustav  Robert  Kirchhoff  ^  (1824- 
1887),  who  was  professor  at  Breslau,  Heidelberg  and  since  1875  at 
Berlin,  investigated  the  distribution  of  a  current  over  a  flat  conductor, 
and  also  the  strength  of  current  in  each  branch'  of  a  network  of  linear 
conductors. 

The  entire  subject  of  electro-magnetism  was  revolutionized  by 
James  Clerk  Maxwell  (1831-1870)).  He  was  born  near  Edinburgh, 
entered  the  University  of  Edinburgh,  and  became  a  pupil  of  Kelland 
and  Forbes.  In  1850  he  went  to  Trinity  College,  Cambridge,  and 
came  out  Second  Wrangler,  E.  Routh  being  Senior  Wrangler.  Max- 
well then  became  lecturer  at  Cambridge,  in  1S56  professor  at  Aber- 
deen, and  in  i860  professor  at  King's  College,  London.  In  1865  he 
retired  to  private  life  until  1S71,  when  he  became  professor  of  physics 
at  Cambridge.  Maxwell  not  only  translated  into  mathematical  lan- 
guage the  experimental  results  of  Michael  Faraday,  but  established 
the  electro-magnetic  theory  of  light,  since  verified  experimentally  by 
H.  R,  Hertz.  His  first  researches  thereon  were  pul:)lished  in  1864. 
In  1 87 1  appeared  his  great  Treatise  on  Eledricily  and  Magnetism. 
He  constructed  the  electro-magnetic  theory  from  general  equations, 
which  are  established  upon  purely  dynamical  principles,  and  which 
determine  the  state  of  the  electric  field.  It  is  a  mathematical  discus- 
sion of  the  stresses  and  strains  in  a  dielectric  medium  subjected  to 
electro-magnetic  forces.  The  electro-magnetic  theory  has  received 
developments  from  Lord  Rayleigh,  J.  J.  Thomson,  H.  A.  Rowland, 
R.  T.  Glazebrook,  H.  Helmholtz,  L.  Boltzmann,  O.  Heaviside,  J.  H. 
Poynting,  and  others.  Hermann  von  Helmholtz  (1821-1894)  was 
born  in  Potsdam,  studied  medicine,  was  assistant  at  the  charity  hos- 
pital in  Berlin,  then  a  military  surgeon,  a  teacher  of  anatomy,  a  pro- 
fessor of  physiology  at  Konigsberg,  at  Bonn  and  at  Heidelberg.  In 
187 1  he  went  to  Berlin  as  successor  to  Magnus  in  the  chair  of  physics. 
In  1887  he  became  director  of  the  new  Physikalisch-Technische 
Reichsanstalt.  As  a  young  man  of  twenty-six  he  published  the  now 
famous  pamphlet  Ueher  die  Erhaltung  dcr  Kraft.  His  work  on  Tonemp- 
findung  was  written  in  Heidelberg.  After  he  went  to  Berlin  he  was 
engaged  chiefly  on  inquiries  in  electricity  and  hydrodynamics.  Helm- 
holtz aimed  to  determine  in  what  direction  experiments  should  be 
made  to  decide  between  the  theories  of  W.  Weber,  F.  E.  Neumann, 
G.  F.  B.  Riemann,  and  R.  Clausius,  who  had  attempted  to  explain 
electro-dynamic  phenomena  by  the  assumption  of  forces  acting  at  a 
distance  between  two  portions  of  the  hypothetical  electrical  fluid,-  - 
1  W.  Voigt,  Znm  Gcdachtniss  von  G.  KirchhoJJ,  Gottingen,  1888. 


M'PLIKl)  MATHEMATICS  475 

the  intensity  l^eing  dependent  not  only  on  tlic  distance,  but  also  on 
the  velocity  and  acceleration,— and  the  theory  of  M.  Faraday  and 
C.  Maxwell,  which  discarded  action  at  a  distance  and  assumed  stresses 
and  strains  in  the  dielectric.  His  experiments  favored  the  British 
theory.  He  wrote  on  abnormal  dispersion,  and  created  analogies 
between  electro-dynamics  and  hydrodynamics.  Lord  Rayleigh  com- 
pared electro-magnetic  problems  with  their  mechanical  analogues, 
gave  a  dynamical  theory  of  diffraction,  and  applied  Laplace's  coeffi- 
cients to  the  theory  of  radiation.  H.  Rowland  made  some  emenda- 
tions on  G.  G.  Stokes'  paper  on  diffraction  and  considered  the  pro- 
pagation of  an  arbitrary  electro-magnetic  disturbance  and  spherical 
waves  of  light.  Electro-magnetic  induction  has  been  investigated 
mathematically  by  Oliver  Heaviside,  and  he  showed  that  in  a  cable 
it  is  an  actual  benefit.  O.  Heaviside  and  J.  H.  Poynting  have  reached 
remarkable  mathematical  results  in  their  interpretation  and  develop- 
ment of  Maxwell's  theory.  Most  of  Heaviside's  papers  have  been 
published  since  1882;  they  cover  a  wide  field. 

One  part  of  the  theory  of  capillary  attraction,  left  defective  by  P.  S. 
Laplace,  namely,  the  action  of  a  solid  upon  a  liquid,  and  the  mutual 
action  between  two  hquids,  was  made  dynamically  perfect  by  K.  F. 
Gauss.  He  stated  the  rule  for  angles  of  contact  between  liquids  and 
solids.  A  similar  rule  for  liquids  was  established  by  Franz  Ernst 
Neumann.  Chief  among  m  )re  recent  workers  on  the  mathematical 
theory  of  capillarity  are  Lord  Rayleigh  and  E.  Mathieu. 

The  great  principle  of  the  conservation  of  energy  was  established 
by  Robert  Mayer  (1814-1S7S),  a  physician  in  Heilbronn,  and  again 
independently  by  Ludwig  A.  Colding  of  Copenhagen,  J.  P.  Joule,  and 
H.  Helmholtz.  James  Prescott  Joule  (181S-1S89)  determined  ex- 
perimentally the  mechanical  equivalent  of  heat.  H.  Helmholtz  in 
1847  applied  the  conceptions  of  the  transformation  and  conservation 
of  energy  to  the  various  branches  of  physics,  and  thereby  linked  to- 
gether many  well-known  phenomena.  These  labors  led  to  the  aban- 
donment of  the  corpuscular  theory  of  heat.  The  mathematical  treat- 
ment of  thermic  problems  was  demanded  by  practical  considerations. 
Thermodynamics  grew  out  of  the  attempt  to  determine  mathemati- 
cally how  much  work  can  be  gotten  out  of  a  steam  engine.  Sadi 
Nicolas  Leonhard  Carnot  (1796-1832)  of  Paris,  an  adherent  of  the 
corpuscular  theory,  gave  the  first  impulse  to  this.  The  principle 
known  by  his  name  was  published  in  1S24.  Though  the  importance 
of  his  work  was  emphasized  by  B.  P.  E.  Clapcyron,  it  did  not  meet 
with  general  recognition  until  it  was  brought  forward  by  William 
Thomson  (Lord  Kelvin).  The  latter  pointed  out  the  necessity  of 
modifying  Carnot's  reasoning  so  as  to  bring  it  into  accord  with  the 
new  theory  of  heat.  William  Thomson  showed  in  1848  that  Carnot's 
principle  led  to  the  conception  of  an  absolute  scale  of  temperature. 
In  1849  he  published  "an  account  of  Carnot's  theory  of  the  motive 


476  A  HISTORY  OF  MATHEMATICS 

power  of  heat,  with  numerical  results  deduced  from  Regnault's  ex- 
periments." In  February,  1850,  Rudolph  Clausius  (1822-1888),  then 
in  Zurich  (afterwards  professor  in  Bonn),  communicated  to  the  Berlin 
Academy  a  paper  on  the  same  subject  which  contains  the  Protean 
second  law  of  thermodynamics.  In  the  same  month  William  John 
M.  Rankine  (1S20-1872),  professor  of  engineering  and  mechanics 
at  Glasgow,  read  before  the  Royal  Society  of  Edinburgh  a  paper  in 
which  he  declares  the  nature  of  heat  to  consist  in  the  rotational  mo- 
tion of  molecules,  and  arrives  at  some  of  the  results  reached  previously 
by  R.  Clausius.  He  does  not  mention  the  second  law  of  thermody- 
namics, but  in  a  subsequent  paper  he  declares  that  it  could  be  derived 
from  equations  contained  in  his  first  paper.  His  proof  of  the  second 
law  is  not  free  from  objections.  In  March,  185 1,  appeared  a  paper 
of  William  Thomson  (Lord  Kelvin)  which  contained  a  perfectly 
rigorous  proof  of  the  second  law.  He  obtained  it  before  he  had  seen 
the  researches  of  R.  Clausius.  The  statement  of  this  law,  as  given  by 
Clausius,  has  been  much  criticised,  particularly  by  W.  J.  M.  Rankine, 
Theodor  Wand,  P.  G.  Tait,  and  Tolver  Preston.  Repeated  efforts  to 
deduce  it  from  general  mechanical  principles  have  remained  fruitless. 
The  science  of  theormodynamics  was  developed  with  great  success 
by  W.  Thomson,  Clausius,  and  Rankine.  As  early  as  1852  W.  Thom- 
son discovered  the  law  of  the  dissipation  of  energy,  deduced  at  a 
later  period  also  by  R.  Clausius.  The  latter  designated  the  non- 
transformable  energy  by  the  name  entropy,  and  then  stated  that  the 
cntrop}^  of  the  universe  tends  toward  a  maximum.  I'^or  entropy 
Rankine  used  the  term  thermodynamic  function.  Thermodynamic 
investigations  have  been  carried  on  also  by  Gustav  Adolph  Hirn  (1815- 
1890)  of  Colmar,  and  H.  Helmholtz  (monocyclic  and  polycyclic  sys- 
tems). Valuable  graphic  methods  for  the  study  of  thermodynamic 
relations  were  devised  by  J.  W.  Gibbs  of  Yale  College. 

Josiah  Willard  Gibbs  (1839-1903)  was  born  in  New  Haven,  Conn., 
and  spent  the  first  five  years  after  graduation  mainly  in  mathematical 
studies  at  Yale.  He  passed  the  winter  of  1866-1867  in  Paris,  of  1867- 
1868  in  Berlin,  of  1868-1869  in  Heidelberg,  studying  physics  and 
mathematics.  In  187 1  he  was  elected  professor  of  mathematical  physics 
at  Yale.  "His  direct  geometrical  or  graphical  bent  is  shown  by  the  at- 
traction which  vectorial  modes  of  notation  in  physical  analysis  exerted 
over  him,  as  they  had  done  in  a  more  moderate  degree  over  C.  Max- 
well." Greatly  influenced  by  Sadi  Carnot,  by  William  Thomson  (Lord 
Kelvin)  and  especially  by  R.  Clausius,  Gibbs  began  in  1873  to  pre- 
pare papers  on  the  graphical  expression  of  thermodynamic  relations, 
in  which  energy  and  entropy  appeared  as  variables.  He  discusses  the 
entropy-temperature  and  entropy-volume  diagrams,  and  the  volume- 
energy-entropy  surface  (described  in  C.  Maxwell's  Theory  of  Heat\ 
Gibbs  formulated  the  energy-entropy  criterion  of  equilibrium  and 
stability,  and  expressed  it  in  a  form  applicable  to  complicated  problems 


APPLIED  MATHEMATICS  477 

of  dissociation.  That  chemistry  has  tended  to  take  a  mathematical 
turn,  says  E.  Picard,  is  evident  from  "the  celebrated  memoir  of  J.  W. 
Gibbs  on  the  equilibrium  of  chemical  systems,  so  analytic  in  char- 
acter, and  where  is  needed  some  effort  on  the  part  of  the  chemists  to 
recognize,  under  their  algebraic  mantle,  laws  of  high  importance." 

In  1902  appeared  J.  W.  Gibbs'  Elementary  Principles  in  Statistical 
Mechanics,  developed  with  special  reference  to  the  rational  foundation 
of  thermodynamics.  The  modern  kinetic  theory  of  gases  was  mainly 
the  work  of  R.  Clausius,  C.  Maxwell,  and  Boltzmann.  "In  reading 
Clausius  we  seem  to  be  reading  mechanics;  in  reading  Maxwell,  and 
in  much  of  L.  Boltzmann's  most  valuable  work,  we  seem  rather  to  be 
reading  in  the  theory  of  probabilities."  C.  Maxwell,  and  L.  Boltzmann 
are  the  creators  of  "statistical  dynamics."  While  they  treated  of 
molecules  of  matter  directly,  J.  W.  Gibbs  considers  "the  statistics 
of  a  delinite  vast  aggregation  of  ideal  similar  mechanical  systems  of 
types  completely  defined  beforehand,  and  then  compares  the  precise 
results  reached  in  this  ideal  discussion  with  the  principles  of  thermo- 
dynamics, already  ascertained  in  the  semi-empirical  manner."  ^  Im- 
portant works  on  thermodynamics  were  prepared  by  R.  Clausius 
in  1875,  by  R.  Riihlmann  in  1875,  and  by  H.  Poincare  in  1892. 

In  the  study  of  the  law  of  dissipation  of  energy  and  the  principle 
of  least  action,  mathematics  and  metaphysics  met  on  common  ground. 
The  doctrine  of  least  action  was  first  propounded  by  P.  L.  M.  Mau- 
pertius  in  1744.  Two  years  later  he  proclaimed  it  to  be  a  universal 
law  of  nature,  and  the  first  scientific  proof  of  the  existence  of  God. 
It  was  weakly  supported  by  him,  violently  attacked  by  Konig  of 
Leipzig,  and  keenly  defended  by  L.  Euler.  J.  Lagrange's  conception 
of  the  principle  of  least  action  became  the  mother  of  analytic  me- 
chanics, but  his  statement  of  it  was  inaccurate,  as  has  been  remarked 
by  Josef  Bertrand  in  the  third  edition  of  the  Mecanique  Analytique. 
The  form  of  the  principle  of  least  action,  as  it  now  exists,  was  given 
by  W.  R.  Hamilton,  and  was  extended  to  electro-dynamics  by  F.  E. 
Neumann,  R.  Clausius,  C.  Maxwell,  and  H.  Helmholtz.  To  sub- 
ordinate the  principle  to  all  reversible  processes,  H.  Helmholtz  intro- 
duced into  it  the  conception  of  the  "kinetic  potential."  In  this  form 
the  principle  has  universal  validity. 

An  offshoot  of  the  mechanical  theory  of  heat  is  the  modem  kinetic 
theory  of  gases,  developed  mathematically  by  R.  Clausius,  C.  Maxwell, 
Ludwig  Boltzmann  of  Vienna,  and  others.  The  first  suggestions  of  a 
kinetic  theory  of  matter  go  back  as  far  as  the  time  of  the  Greeks.  The 
earliest  work  to  be  mentioned  here  is  that  of  Daniel  Bernoulli,  1738. 
He  attributed  to  gas-molecules  great  velocity,  explained  the  pressure 
of  a  gas  by  molecular  bombardment,  and  deduced  Boyle's  law  as  a 
consequence  of  his  assumptions.  Over  a  century  later  his  ideas  were 
taken  up  by  J.  P.  Joule  (in  1846),  A.  K.  Kronig  (in  1S56),  and  R. 
*  Frocecd.  of  the  Royal  Soc.  of  London,  Vol    75.  1905,  p.  293. 


47S  A  HISTORY  OF  MATHEMATICS 

Clausius  (in  1857).  J.  P.  Joule  dropped  his  speculations  on  this 
subject  when  he  began  his  experimental  work  on  heat.  A.  K.  Kronig 
explained  by  the  kinetic  theory  the  fact  determined  experimentally 
by  Joule  that  the  internal  energy  of  a  gas  is  not  altered  by  expansion 
when  no  external  work  is  done.  R.  Clausius  took  an  important  step 
in  supposing  that  molecules  may  have  rotary  motion,  and  that  atoms 
in  a  molecule  may  move  relatively  to  each  other.  He  assumed  that 
the  force  acting  between  molecules  is  a  function  of  their  distances, 
that  temperature  depends  solely  upon  the  kinetic  energy  of  molecular 
motions,  and  that  the  number  of  molecules  which  at  any  moment  are 
so  near  to  each  other  that  they  perceptibly  influence  each  other  is 
comparatively  so  small  that  it  may  be  neglected.  He  calculated  the 
average  velocities  of  molecules,  and  explained  evaporation.  Objections 
to  his  theory,  raised  by  C.  H.  D.  Buy's-Ballot  and  b}'  Emil  Jochmann, 
were  satisfactorily  answered  by  R.  Clausius  and  C.  Maxwell,  except  in 
one  case  where  an  additional  hypothesis  had  to  be  made.  C.  Maxwell 
proposed  to  himself  the  problem  to  determine  the  average  number 
of  molecules,  the  velocities  of  which  lie  between  given  hmits.  His 
expression  therefor  constitutes  the  important  law  of  distribution  of 
velocities  named  after  him.  By  this  law  the  distribution  of  molecules 
according  to  their  velocities  is  determined  by  the  same  formula  (given 
in  the  theory  of  probability)  as  the  distribution  of  empirical  observa- 
tions according  to  the  magnitude  of  their  errors.  The  average  mo- 
lecular velocity  as  deduced  by  C.  Maxwell  differs  from  that  of  R. 
Clausius  by  a  constant  factor.  C.  Maxwell's  first  deduction  of  this 
average  from  his  law  of  distribution  was  not  rigorous.  A  sound  deriva- 
tion was  given  by  O.  E.  Meyer  in  1866.  C.  Maxwell  predicted  that 
so  long  as  Boyle's  law  is  true,  the  coefficient  of  viscosity  and  the  coeffi- 
cient of  thermal  conductivity  remain  independent  of  the  pressure. 
His  deduction  that  the  coefficient  of  viscosity  should  be  proportional 
to  the  square  root  of  the  absolute  temperature  appeared  to  be  at 
variance  with  results  obtained  from  pendulum  experiments.  This 
induced  him  to  alter  the  very  foundation  of  his  kinetic  theory  of  gases 
by  assuming  between  the  molecules  a  repelling  force  varying  inversely 
as  the  fifth  power  of  their  distances.  The  founders  of  the  kinetic 
theory  had  assumed  the  molecules  of  a  gas  to  be  hard  elastic  spheres; 
but  Maxwell,  in  his  second  presentation  of  the  theory  in  1866,  went 
on  the  assumption  that  the  molecules  behave  like  centres  of  forces. 
He  demonstrated  anew  the  law  of  distribution  of  velocities;  but  the 
proof  had  a  flaw  in  argument,  pointed  out  by  L.  Boltzmann,  and 
recognized  by  C.  Maxwell,  who  adopted  a  somewhat  different  form 
of  the  distributive  function  in  a  paper  of  1879,  intended  to  explain 
mathematically  the  effects  observed  in  Crookes'  radiometer.  L.  Boltz- 
mann gave  a  rigorous  general  proof  of  Maxwell's  law  of  the  distribu- 
tion of  velocities. 
N:>  le  of  the  fundamental  assumptions  in  the  kinetic  theory  of  gaser* 


APPLIED  MATHEMATICS  479 

leads  by  the  laws  of  probability  to  results  in  very  close  agreement 
with  observation.  L.  Boltzmann  tried  to  establish  kinetic  theories 
of  gases  by  assuming  the  forces  between  molecules  to  act  according  to 
different  laws  from  those  previously  assumed.  R.  Clausius,  C.  Max- 
well, and  their  j^redecessors  took  the  mutual  action  of  molecules  in 
collision  as  repulsive,  but  L.  Boltzmann  assumed  that  they  may  be 
attractive.  Experiments  of  J.  P.  Joule  and  Lord  Kelvin  seem  to  sup- 
port the  latter  assumption. 

Among  the  later  researches  on  the  kinetic  theory  is  Lord  Kelvin's 
disproof  of  a  general  theorem  of  C.  Maxwell  and  L.  Boltzmann,  as- 
serting that  the  average  kinetic  energy  of  two  given  portions  of  a 
system  must  be  in  the  ratio  of  the  number  of  degrees  of  freedom  of 
those  portions. 

In  recent  years  the  kinetic  theory  of  gases  has  received  less  attention; 
it  is  considered  inadequate  since  the  founding  of  the  quantum  hypothe- 
sis in  physics. 

Relativity 

Profound  and  startling  is  the  "  theory  of  relativity."  On  the  theory 
that  the  ether  was  stationary  it  was  predicted  that  the  time  required 
for  light  to  travel  a  given  distance  forward  and  back  would  be  different 
when  the  path  of  the  light  was  parallel  to  the  motion  of  the  earth  in  its 
orbit  from  what  it  was  when  the  path  of  the  light  was  perpendicular. 
In  1887  A.  A.  Michelson  and  E.  W.  Morely  found  experimentallythat 
such  a  difference  in  time  did  not  exist.  More  generally,  the  results 
of  this  and  other  experiments  indicate  that  the  earth's  motion  through 
space  cannot  be  detected  by  observations  made  on  the  earth  alone. 
In  order  to  explain  Michelson  and  Morley's  negative  result  and  at 
the  same  time  save  the  stationary-ether  theory,  H.  A.  Lorentz  con- 
structed in  1895  a  "contraction  h>qDOthesis,"  according  to  which  a  mov- 
ing solid  contracts  slightly  longitudionally.  This  same  idea  occurred 
independently  to  G.  F.  Fitzgerald.  In  1904  and  in  his  Columbia  Uni- 
versity Lectures  Lorentz  aimed  to  reduce  the  electromagnetic  equations 
for  a  moving  system  to  the  form  of  those  that  hold  for  a  system  at 
rest.    Instead  of  x,  y,  z,  t  he  introduced  new  independent  variables, 

viz.,  x'=\'y{x—vt),  y=\y,  z'=\z,  t'=\y(t—-^x),  where  7  depends 

upon  velocity  of  light  c  and  of  the  moving  body  v,  and  X  is  a  numerical 
coefficient  such  that,  X=i  when  v=o.  His  fundamental  equations 
turned  out  to  be  invariant  under  this  now  called  "Lorentz  transforma- 
tion." In  1906  H.  Poincare  made  use  of  this  transformation  for  the 
treatment  of  the  dynamics  of  the  electron  and  also  of  universal  gravi- 
tation.^ In  1905  A.  Einstein  published  a  paper  on  the  electrodynam- 
ics of  moving  bodies  in  Annalen  der  Physik,  Vol.  17,  aiming  at  perfect 

1  L.  Silberstein,  The  Theory  of  Relalivily,  London,  19 14,  p.  87. 


48o  A  HISTORY  OF  MATHEMATICS 

reciprocity  or  eciuivalence  of  a  pair  of  moving  systems,  and  investi- 
gating the  whole  problem  from  the  bottom,  carefully  considering  the 
matter  of  "simultaneous"  events  in  two  distant  places;  he  has  suc- 
ceeded in  giving  plausible  support  to  and  a  striking  interpretation 
of  Lorentz's  transformations.  Einstein  opened  the  way  to  the  modern 
"theory  of  relativity."  He  developed  it  somewhat  more  fully  in  1907. 
A  fundamental  point  of  view  in  his  theory  was  that  mass  and  energy 
are  proportional.  For  the  purpose  of  taking  account  of  gravitational 
phenomena,  Einstein  generalized  his  theory  by  assuming  that  mass 
and  weight  are  also  proportional,  so  that,  for  example,  a  ray  of  light 
is  attracted  by  matter.  The  mathematical  part  of  Einstein's  theory, 
as  developed  by  M.  Grossmann  in  1913,  employs  quadratic  differential 
forms  and  the  absolute  calculus  of  Gregorio  Ricci  of  Padua.  Another 
remarkable  speculation  was  brought  out  in  1908  by  Hermann  Min- 
kowski who  read  a  lecture  on  Raum  und  Zeit,  in  which  he  maintained 
that  the  new  views  of  space  and  time,  developed  from  experimental 
considerations,  are  such  that  "space  by  itself  and  time  by  itself  sink 
into  the  shadow  and  only  a  kind  of  union  of  the  two  retains  self-de- 
pendence." No  one  notices  a  place,  except  at  some  particular  time, 
nor  time  except  at  a  particular  place.  A  system  of  values  x,  y,  z,  t 
he  calls  a  "world  point"  (Weltpunkt) ;  the  life-path  of  a  material  point 
in  four-dimensional  space  is  a  "world  line."  The  idea  of  time  as  a 
fourth  dimension  had  been  conceived  much  earlier  by  J.  Lagrange  in 
his  Thcorie  dcs  foncUons  analytiques  and  by  D'Alembert  in  his  article 
"Dimension"  in  Diderot's  Encychpedie,^  i754-  H.  Minkowski  con- 
siders the  group  belonging  to  the  differential  equation  for  the  propaga- 
tion of  waves  of  light.  Hermann  Minkowski  (1864-1909),  was  bom  at 
Alexoten  in  Russia,  studied  at  Konigsberg  and  Berlin,  held  associate 
professorships  at  Bonn  and  Konigsberg  and  was  promoted  to  a  full 
professorship  at  Konigsberg  in  1895.  In  1896  he  went  to  the  poly- 
technic school  at  Zurich  and  in  1903  to  Gottingen.  The  importance 
which  H.  Minkowski,  starting  with  the  principle  of  relativity  in  the 
form  given  it  by  Einstein,  has  given  to  the  Lorentzian  transforma- 
tions by  the  introduction  of  a  four  dimensional  manifoldness  or  space- 
time-world,  has  been  made  intuitively  evident  by  a  nimiber  of  writers, 
particularly  F.  Klein  (1910),  L.  Heffter  (191 2),  A.  Brill  (191 2),  and 
H.  E.  Timerding  (1912).  F.  Klein  said:  "What  the  modem  physicists 
call  '  theory  of  relativity '  is  the  theory  of  invariants  of  the  fourth  di- 
mensional space-time-region  x,  v,  2,  /  (Minkowski's  world)  in  relation 
to  a  definite  group  of  collineations,  namely  the  'Lorentz-group.'  "  ^ 
A  novel  presentation  aiming  at  great  precision  was  given  in  1914  by 
Alfred  A.  Robb  who  on  the  idea  of  "conical  order"  and  21  postulates 
builds  up  a  system  in  which  the  theory  of  space  becomes  absorbed  in 
the  theory  of  time.    A  philosophical  discussion  of  relativity,  mechan- 

^  R.  C.  Archibald  in  Bull.  Am.  Math.  Soc,  Vol.  20,  1Q14,  p.  410. 
2  Klein  in  Jahresb.  d.  d.  Math.  Verein.,  Vol.  19,  1910,  p.  287. 


APPLIED  MATHEMATICS  481 

ics  and  geometrical  axioms  is  given  by  Federigo  Enriques  of  Bologna 
in  his  Problems  of  Science  (1906),  which  has  been  translated  into  Eng- 
lish by  Katharine  Royce  in  1914.  F.  Enriques  argues  that  certain 
optical  and  electro-optical  phenomena  seem  to  lead  to  a  direct  contra- 
diction of  the  principles  of  classic  mechanics,  especially  of  Newton's 
principle  of  action  and  reaction.  "Physics,"  says  Enriques,  "instead 
of  affording  a  more  precise  verification  of  the  classic  mechanics,  leads 
rather  to  a  correction  of  the  principles  of  the  latter  science,  taken  a 
priori  as  rigid." 

The  Russian  mathematician,  Vladimir  Varicak  found  that  the 
Lobachevskian  geometry  presented  itself  as  the  best  adapted  for  the 
mathematical  treatment  of  the  physics  of  relativity.  He  enters  upon 
optical  phenomena  and  the  resolution  of  paradoxes  due  to  Ehrenfest 
and  Bonn.  Starting  from  this  point  of  ^-iew,  E.  Borel  in  1913  was  able 
to  deduce  new  consequences  of  the  theory  of  relativity.  One  advantage 
of  Varicak's  presentation  is  that  it  safeguards  the  parallelism  between 
the  old  enunciations  of  physics  and  the  new.  L.  Rougier  of  Lyon  ^ 
asks  the  question,  is  then  the  Lobachevskian  geometry  physically 
true  and  the  Euclidean  wrong?  No.  One  may  keep,  says  he,  the  or- 
dinary geometry'  for  the  discussion  of  the  physics  of  relati\dty,  as  is 
done  by  H.  A.  Lorentz  and  A.  Einstein,  or  one  may  add  a  fourth  imag- 
inary dimension  to  our  three  dimensions  in  the  manner  of  H.  Min- 
kowski, or  one  may  use  the  non-Euclidean  geometries  of  mechanics 
and  electromagnetics  developed  by  E.  B.  Wilson  and  G.  N.  Lewis,- 
then  of  Boston.  Each  of  these  interpretations  enjoys  some  particular 
advantages  of  its  own. 

Nomography 

The  use  of  simple  graphic  tables  for  computation  is  encountered  in 
antiquity  and  the  middle  ages.  The  graphic  solution  of  spherical 
triangles  was  in  vogue  in  the  time  of  Hipparchus,^  and  in  the  seven- 
teenth century  by  W.  Oughtred,^  for  instance.  Edmund  Wingate's 
Construction  and  Use  of  the  Line  of  Proportion,  London,  1628,  de- 
scribed a  double  scale  upon  which  numbers  are  indicated  by  spaces 
on  one  side  of  a  straight  line  and  the  corresponding  logarithms  by 
spaces  on  the  other  side  of  the  line.^  Recently  this  idea  has  been  car- 
ried out  by  A.  Tichy  in  his  Graphische  Logarithmentafeln,  Vienna,  1897. 
The  Longitude  Tables  and  Horary  Tables  of  Margetts,  London,  1791, 
were  graphical.  More  systematic  use  of  this  idea  was  made  by 
Pouchet  in  his  Arilhmetiqiie  lineaire,  Rouen,  1795.  In  1842  appeared 
the  A  namorpJwse  logarithmique  of  the  Parisian  engineer  Leon  Lalanne 

'  L'Enseignement  Mathematigue,  Vol.  XVI,  19 14,  p.  17. 
'  Proceed.  Am.  Acad,  of  Arts  and  Science,  Vol.  48,  191 2. 
'  A.  von  Braunmiihl,  Ceschichte  der  Trigonometric,  Leipzig,  Vol.  I,  1900,  pp.  3,  iO| 

^5'  191. 

*  F.  Cajori,  William  Otightrcd,  Chicago  and  London,  1916. 

*  F.  Cajori,  Colorado  College  Publication,  General  Series  47,  19 10,  p.  182. 


482  A  HISIORV  OF  MATHEMATICS 

(1811-1892)  in  which  the  distances  of  points  from  the  origin  are  not 
necessarily  proportional  to  the  actual  values  of  the  data,  but  may  be 
other  functions  of  them,  judiciously  chosen.  In  the  product  2122=23, 
the  variables  21  and  22  are  brought  in  correspondence,  respectively, 
with  the  straight  lines  x=log  21,  >'=log  29,  so  that  x+y=\og  23, 
which  represents  the  straight  lines  peri:>endicular  to  the  bisectors  of  the 
angle  between  the  co-ordinate  axes.  Advances  along  this  line  were 
made  by  J.  Massau  of  the  University  of  Ghent,  in  18S4,  and  E.  A. 
Lallemand  in  1886.  The  Scotch  Captain  Patrick  Weir  in  1889  gave 
an  azimuth  diagram  which  was  an  anticipation  of  a  spherical  triangle 
nomogram.  But  the  real  creator  of  nomogr-aphy  is  Maurice  d'Ocagne 
of  the  Ecole  Polytechnique  in  Paris,  whose  first  researches  appeared  in 
1891;  his  Traiie  de  nomographie  came  out  in  1S99.  The  principle  of 
anamorphosis,  by  successive  generalizations,  "has  led  to  the  con- 
sideration of  equations  representable  not  only  by  two  systems  of 
straight  lines  parallel  to  the  axes  of  co-ordinates  and  one  other  un- 
restricted system  of  straight  lines,  but  by  three  systems  of  straight 
lines  under  no  such  restrictions."  D'Ocagne  also  studied  equations 
representable  by  means  of  systems  of  circles.  He  has  introduced  the 
method  of  coUinear  points  by  which  "it  has  been  possible  to  represent 
nomographically  equations  of  more  than  three  variables,  of  which 
the  previous  methods  gave  no  convenient  representation."  ^ 

Mathematical  Tables 

The  increased  accuracy  now  attainable  in  astronomical  and  geodetic 
measurements  and  the  desire  to  secure  more  complete  elimination  of 
errors  from  logarithmic  tables,  has  led  to  recomputations  of  logarithms. 
Edward  Sang  of  Edinburgh  published  in  1S71  a  7-place  table  of  com- 
mon logarithms  of  numbers  to  200,000.  These  were  mainly  derived 
from  his  unpublished  28-place  table  of  logarithms  of  primes  to  10,037 
and  composite  numbers  to  20,000,  and  his  15-place  table  from  100,000 
to  370,000.^  In  1889  the  Geographical  Institute  of  Florence  issued  a 
photographic  reproduction  of  G.  F.  Vega's  Thesaurus  of  1794  (10 
figures).  Vega  had  computed  A.  Vlacq's  tables  anew,  but  his  last 
figure  was  unreliable.  In  1891  the  French  Government  issued  8- 
place  tables  which  were  derived  from  the  unpublished  Tables  du 
Cadastre  (14-places,  12  correct)  which  had  been  computed  near  the 
close  of  the  eighteenth  century  under  the  supervision  of  G.  Riche  de 
Prony.  These  tables  give  logarithms  of  numbers  to  120,000,  and  of 
sines  and  tangents  for  every  10  centesimal  seconds,  the  quadrant  being 
divided  centesimally.^    Prony  consulted  A.  M.  Legendre  and  other 

1  D'Ocagne  in  Napier  Tercentenary  Memorial  Volume,  London,  1915,  pp.  279- 
283.    See  also  D'Ocagne,  Le  calcid  slmplifie,  Paris,  IQ05,  pp.  145-153. 

-  E.  M.  Horsburgh,  Napier  Tercentenary  Cclebralion  Ihmdbook,  10 14,  pp.  38-43. 

*This  and  similar  information  is  drawn  from  J.  W.  L.  Claisher  in  Napier  Ter- 
centenary Memorial  Volume,  London,  1915,  pp.  71-/3. 


APPLIED  MATHEMAIICS  483 

inathematicians  on  the  choice  of  methods  and  formulas,  and  cntrustec 
the  computation  of  primary  results  to  professional  calculators,  while 
the  task  of  filling  the  rest  of  the  columns  beyond  the  primary  results 
was  performed  by  assistants  "apt  merely  in  perfornaing  additions" 
by  the  use  of  the  method  of  differences.  "  It  is  curious,"  says  D'Ocagne 
"to  note  that  the  majority  of  these  assistants  had  been  recruited  from 
among  the  hair-dressers  whom  the  abandonment  of  the  powdered  wig 
in  men's  fashion  had  deprived  of  a  livelihood." 

In  1S91  M.  J.  dc  M endizabcl-Tamborrel  published  at  Paris  tables  of 
logarithms  of  numbers  to  125,000  (8  places)  and  of  sines  and  tangents 
(7  or  8  places)  for  every  millionth  of  the  circumference,  which  were 
almost  wholly  derived  from  original  lo-place  calculations.  W.  W. 
Diiffield  pubHshed  in  the  Report  of  the  U.  S.  Coast  and  Geodetic  Sur- 
vey, 1 895-1 896,  a  ID-place  table  of  logarithms  of  numbers  to  100,000, 
in  1910,  8-place  tables  of  numbers  to  200,000  and  trigonometric  tables 
to  every  sexagesimal  second  w-ere  published  by  /.  Bauschinger  and 
/.  Peters  of  Strassburg.  A  special  machine  was  constructed  for  the 
computation  of  these  tables.  In  191 1  //.  Andoyer  of  Paris  published  a 
14-place  table  of  logarithms  of  sines  and  tangents  to  every  10  sex- 
agesimal seconds.  "This  table  was  derived  from  a  complete  recalcula- 
tion, made  entirely  by  M,  Andoyer  himself,  without  any  assistance, 
personal  or  mechanical." 

In  recent  years  a  demand  has  arisen  for  tables  giving  the  natural 
values  of  sines  and  cosines.  In  191 1  /.  Peters  published  in  Berlin  such 
a  table,  extending  from  0°  to  90°,  and  carried  to  21  decimals,  for  every 
10  sexagesimal  seconds  (and  for  every  second  of  the  first  six  degrees). 
Extensive  tables  of  natural  values,  first  computed  by  Rhaeticus  and 
])ublished  in  1613,  were  abandoned  after  the  invention  of  logarithms, 
but  are  now  returning  in  use  again,  since  they  are  better  fitted  for  the 
growing  practice  of  calculating  directly  by  means  of  machines  and 
without  resort  to  logarithms. 

The  decimal  division  of  angles  has  been  agitated  again  in  recent 
years.  In  1900  R.  Mehmke  made  a  report  to  the  German  Mathema- 
tiker  Vereinigung.^  Why  are  degrees  preferred  to  radians  in  practical 
trigonometry?  Because,  on  account  of  the  periodicity  of  the  trig- 
onometric functions,  we  frequently  would  have  to  add  and  subtract 
TT  or  2  7r  which  are  irrational  numbers  and  therefore  objectionable. 
The  sexagesimal  subdivision  of  the  degree  which  resulted  in  great 
hannony  among  the  Babylonians  who  used  the  sexagesimal  notation 
of  numbers  and  fractions,  and  the  sexagesimal  di\'isions  of  the  day, 
hour  and  minute,  is  less  desirable  now  that  we  have  the  decimal 
notation  of  numbers.  There  has  been  some  difference  of  opinion 
among  advocates  of  the  decimal  system  in  angular  measurement,  what 
unit  should  be  chosen  for  the  decimal  subdivision.     In  1864  Yvon 

1  See  Jahresb.  d.  d.  Math.  Vereinigung,  Leipzig,  Vol.  8,  Part  i,  1900  p.  13Q. 


484  A  HISTORY  OF  MATHEMATICS 

Villarceau,  at  a  meeting  of  the  Bureau  of  Longitudes  in  Paris,  sug- 
gested the  decimal  subdivision  of  the  entire  circumference,  while  in 
1896  Bouquet  de  la  Grye  preferred  the  semi-circumference.  R.  Mehmke 
argues  that  whatever  the  unit  may  be  that  is  subdivided,  the  four 
arithmetical  operations  with  angles  would  be  materially  simplified, 
interpolation  in  the  use  of  trigonometric  tables  would  be  easier,  the 
computation  of  the  lengths  of  arcs  would  be  shorter.  If  the  right 
angle  is  the  unit  that  is  subdivided,  then  the  reduction  of  large  angles 
to  corresponding  acute  angles  can  be  effected  merely  by  the  subtrac- 
tion of  the  integers  i,  2,  3,  .  .  The  determination  of  supplementary  or 
complementary  angles  is  less  laborious,  A  more  convenient  arrange- 
ment of  trigonometric  tables  was  claimed  by  G.  J.  Holiel  and  greater 
comfort  in  taking  observations  was  promised  by  J.  Delambre.  Never- 
theless, no  decimal  division  of  angles  is  at  the  present  time  threatened 
with  adoption,  not  even  in  France. 

A  very  specialized  kind  of  logarithms,  the  so-called  "  Gaussian  loga- 
rithms," which  give  log  {a-\-b)  and  log  (a—b),  when  log  a  and  log  b 
are  known,  were  first  suggested  by  the  Italian  physicist  Guiseppe 
Zecchini  Leonelli  (1776-1847)  in  his  Theorie  des  logarithmes,  Bordeaux 
1803;  the  first  table  was  published  by  K.  F.  Gauss  in  181 2  in  ZacJi's 
MonatUche  Korrespondenz.  It  is  a  5-place  table.  More  recent  tables 
are  the  6-place  tables  of  Carl  Bremiker  (1804-1877)  of  the  geodetic 
institute  of  Berlin,  Siegmund  Gundelfinger  (1846-1910)  of  Darmstadt, 
and  George  William  Jones  (1837-1911)  of  Cornell  tjniversity,  and  the 
7-place  table  of  T.  Wittstein. 

Proceeding  to  hyperbolic  and  exponential  functions,  we  mention 
the  7-place  tables  of  log  10  sink  x  and  log  10  cosh  x  prepared  by  Christoph 
Gudermann  of  Miinster  in  1832,  the  5-place  tables  by  Wilhelm 
Ligowski  (1821-1893)  of  Kiel  in  1890,  the  5-place  tables  by  G.  F. 
Becker  and  C.  E.  Van  Orstrand  in  their  Smithsonian  Mathematical 
Tables,  1909.  Tables  for  sinh  x  and  cosh  x  were  published  by  Ligowski 
(1890),  Burrau  (1907),  Dale,  Becker,  and  Van  Orstrand.  In  the  Cam- 
bridge Philosophical  Transactions,  Vol.  13,  1883,  there  are  tables  for 
log  e'  and  e^  by  J.  W.  L.  Glaisher,  for  e~^  by  F.  W.  Newman, 
G.  F.  Becker  and  C.  E.  Van  Orstrand  also  give  tables  for  these 
functions. 

An  isolated  matter  of  interest  is  the  origin  of  the  term  "radian," 
used  with  trigonometric  functions.  It  first  appeared  in  print  on 
June  5,  1873,  in  examination  questions  set  by  James  Thomson  at 
Queen's  College,  Belfast.  James  Thomson  was  a  brother  of  Lord 
Kelvin.  He  used  the  term  as  early  as  1S71,  while  in  1869  Thomas 
Muir,  then  of  St.  Andrew's  University,  hesitated  between  "rad," 
"radial"  and  "radian."  In  1874  T.  Muir  adopted  "radian"  after 
a  consultation  with  James  Thomson.^ 

'  Nature,  Vol.  83,  pp.  156,  217,  459,  460. 


APPLIED  MATHEiMATICS  485 

Calculating  Machines,  Planimeters,  Integraphs 

The  earliest  calculating  machine,  invented  by  Blaise  Pascal  in  1641, 
was  designed  only  to  effect  addition.  Three  models  of  Pascal's  ma- 
chine are  kept  in  the  Conservatoire  des  arts  et  metiers  in  Paris.  It 
was  G.  W.  Leibniz  who  conceived  the  idea  of  adapting  to  a  machine 
of  this  sort  a  mechanism  capable  of  repeating  several  times,  rapidly, 
the  addition  of  one  and  the  same  number,  so  as  to  effect  multiplication 
mechanically.  Of  the  two  Leibniz  machines  said  to  have  been  con- 
structed, one  (completed  1694?)  is  preserved  in  the  library  of  Hanover. 
This  idea  was  re-invented  and  worked  out  for  actual  use  in  practice 
in  1S20  by  Ch.  X,.  Thomas  de  Colmar  in  his  Arithmometre.  More 
limited  practical  use  was  given  to  the  machine  of  Ph.  M.  Hahn,  first 
constructed  in  Stuttgart  in  1774. 

A  machine  effecting  multiplication,  not  by  repeated  additions,  but 
directly  by  the  multiplication  table,  was  tirst  exhibited  at  the  universal 
exposition  in  Paris  in  18S7.  This  decidedly  original  design  was  the 
invention  of  a  young  Frenchman,  Leon  Bollee,  who  took  also  a  prom- 
inent part  in  the  development  of  the  automobile.  In  his  computing 
machine  there  are  calculating  plates  furnished  with  tongues  of  appro- 
priate length  which  constitute  a  kind  of  multiplication  table,  acting 
directly  on  the  recording  apparatus  of  the  machine.  K  somewhat 
simpler  elaboration  of  the  same  idea  is  due  to  O.  Steiger  (1892)  in  a 
machine  called  the  millionnaire}  In  1892  a  Russian  engineer,  W.  T. 
Odhner,  invented  and  constructed  a  widely  used  machine,  called  the 
Brunsiiga  Calculator,  which  is  of  the  "pin  wheel  and  cam  disc"  type, 
the  first  idea  of  which  goes  back  to  Polenus  (1709)  and  to  Leibniz. 
Of  American  origin  are  the  Burroughs  Adding  and  Listing  machine, 
and  the  Comptometer  invented  about  1SS7  by  Dorr  E.  Felt  of  Chicago. 

The  first  idea  of  automatic  engines  calculating  by  the  aid  of  func- 
tional differences  of  various  orders  goes  back  to  J.  H.  Miiller  (1786), 
but  no  steps  towards  definite  plans  and  actual  construction  were 
taken  before  the  time  of  Babbage.  Charles  Babbage  (1792-1871)  in- 
vented a  machine,  called  a  "difference-engine,"  about  1S12.  Its  con- 
struction was  begun  in  1822  and  was  continued  for  20  years.  The 
British  Government  contributed  £17.000  and  Babbage  himself  £6000. 
Through  some  misunderstanding  with  the  Government,  work  on  the 
engine,  though  nearly  finished,  was  stopped.  Inspired  by  Ch.  Bab- 
bage's  design,  Georg  und  Eduard  Scheutz  (father  and  son)  of  Stock- 
hobn  made  a  difference  engine  which  was  acquired  by  the  Dudley 
Observatory  in  Albany, 

In  1833  Ch.  Babbage  began  the  design  of  his  "analytical  engine"; 

^  D'Ocagne  in  the  Napier  Tercentenary  Memorial  Volume,  London,  1915,  pp.  283- 
285.  For  details,  see  D'Ocaj^ne,  Le  calciil  simpUfie,  Paris,  1905,  pp.  24-92;  Ency- 
kiopMied.  Math.  Wiss.,  Bd.  I,  Leipzig,  1898-1904,  pi-.  952-982;  E.  H.  Horsburgh, 
Napier  Tercentenary  Celebration  Handbook,  Edinburgh,  19 14,  "Calculating  Ma« 
:liines"  by  J.  W  Whipple,  pp.  69-135. 


|cS6  A  HISTORY  OF  MATHEMATICS 

a  small  portion  of  it  was  put  together  before  his  death.  This  engine 
was  intended  to  evaluate  any  algebraic  formula,  for  any  given  values 
of  the  variables.  In  1906  H.  P.  Babbage,  a  son  of  Charles  Babbage, 
completed  part  of  the  engine,  and  a  table  of  25  multiples  of  tt  to  29 
figures  was  published  as  a  specimen  of  its  work.^ 

Planimeters  have  been  designed  independently  and  in  many  dif- 
ferent ways.  It  is  probable  that  J.  M.  Hermann  designed  one  in  1814. 
Planimeters  were  devised  in  1824  by  Gonella  in  Florence,  about  1827 
by  Johannes  Oppikoffcr  (17S3-1859)  -  of  Bern  and  constructed  by 
Ernst  in  Paris,  about  1849  by  WetU  of  Vienna  and  improved  by  the 
astronomer  Peter  Andreas  Hansen  of  Gotha,  about  185 1  by  Edward 
Sang  of  Edinburgh  and  improved  by  Clerk  Maxwell,  J.  Thomson  and 
Lord  Kelvin.  All  of  these  were  rotation  planimeters.  Most  noted  of 
polar  planimeters  are  that  of  Jakob  Amsler  (1823-1912)  and  those  con- 
structed by  Coradi  of  Zurich.  J.  Amsler  was  at  one  time  privatdocent 
at  the  University  of  Zurich,  later  manufacturer  of  instruments  for 
precise  measurements.  He  invented  his  polar  planimeter  in  1854;  his 
account  of  it  was  pubHshed  in  1856. 

Another  interesting  class  of  instruments,  called  "integraphs"  has 
been  invented  by  Abdank  Abakanovicz  (1852-1900)  in  1878  and  by  C. 
Vernon  Boys  ^  in  1882.  These  instruments  draw  an  "integral  curve" 
when  a  pointer  is  passed  round  the  periphery  of  a  figure  whose  area  is 
required.  More  recently  numerous  integraphs  have  been  invented 
through  the  researches  of  E.  Pascal  of  the  University  of  Naples.  Thus 
in  191 1  he  designed  a  polar  integraph  for  the  quadrature  of  differential 
equations. 

*  Napier  Tercentenary  Celebration  Handbook,  19 14,  p.  127. 

2  Morin,  Les  Appareils  d' Integration,  1Q13.    See  E.  M.  Horsburgh,  op.  cit.,  p.  190. 

^  Boys  in  Phil.  Mag.,  1882;  Abdank  Abakanowicz,  Les  Integraphes,  Paris,  1886. 
See  also  H.  S.  Hele  Shaw,  "Graphic  Methods  in  Mechanical  Science"  in  Report  oj 
British  Ass'n  for  1892,  pp.  373-531;  E.  M.  Horsburgh,  Handbook,  pp.  194-206. 


INDEX 


Abacus,  7,  52,  53,  64,  65,  67,  68,  78,  no,  114, 
116,  118-122 

Abbati,  P.,  352;  253 

Abbatt,  R.,  370 

Abdank  Abakanovicz,  486 

Abel,  N.  II.,  410-417;  238,  266,  278,  28g, 
2go,  313,  316,  349-35.>.  374,  3Q.?,  405.  417, 
426,  442;  Abt-lian  functions,  2S1,  313,  342, 
390,  411,  412,  415,  418,  419,  421,  423,  424, 
428,  429;  Abelian  groups,  357,  358,  360, 
388;  Abel's  theorem,  413,  415 

Abraham,  M.,  335 

Abraham,  ibn  Esra,  no 

Abscissa,  175 

Abu  Kamil,  103,  104,  121 

Abu'l  Hasan  Ali,  109 

Abu'I  Jud,  106,  107 

Abu'l  Wefa,  105;  104,  106,  107,  109 

Acceleration,  172,  183 

"Achilles,"  23,  1S2 

Ada  eruditorum,  founded,  209 

Adam,  Ch.,  177 

Adams,  J.  C,  449;  200,  332 

Adler,  A.,  268 

Adrain,  R.,  382 

Aganis,  48,  184 

Aggregates,  theory  of,  24,  66,  172,  325,  326, 
396-406;  Enumerable,  66,  400.  See 
Point-sets 

Agnesi,  M.  G.,  250;  witch,  250 

Agrimensores,  66 

Ahmes,  7,  9-14,  16,  44,  60,  103,  123 

Ahrens,  W.,  323 

Aida  Ammel,  81 

Airy,  J.  B.,  449;  383,  462 

Ajima  Chokuyen,  8i 

Aklimim  papyrus,  14 

Al-Battani,  105;  118 

Albertus  Magnus,  127 

Al-Biruni,  100,  loi,  105,  106 

Alchazin,  107 

.\lcuin,  114;  116,  120 

Alembert,  Jean  le  Rond  d'.  See  D'Alembert 

Alexander,  C.  A.,  266 

Alexander  the  Great,  7 

Alexandrian  School,  first,  29-45;  second,  45- 
52 

Alfonso  X,  iig 

\lgebra,  Arabic,  102-104,  106,  120.  Chinese, 


75,  75.  Egyptian,  13,  14-  Greek,  56- 
62.  Hindu,  93-g6,  103.  Japanese,  79 
Laws  of,  273.  Linear  associative,  285, 
338,  339.  Modern  developments,  329- 
366.  Multiple,  289,  338,  339,  344.  Of 
Renaissance,  137-141.  Origin  of  word, 
103.  Rhetorical,  syncopated  and  sym- 
bolic, in;  the  science  of  time,  sis-  -See 
Ausdehnungslehre,  Covariants,  Equa- 
tions, Invariants,  Quaternions 

Algebra  of  logic,  285 

Algorithm,  119.    Origin  of  word,  102 

Al-Hasan  ibn  Al-Haitam,  104,  107 

Al-Kalsadi,  no,  in 

Al-Karkhi,  106,  107 

Al-Kashi,  108 

Al-Khojandi,  106 

Al-Khowarizmi,  102-104;  94,  106,  108,  118, 
ng 

Al-Kuhi,  105-107 

Allman,  G.  J.,  16;  30 

Almagest,  see  Ptolemy 

Al-Mahani,  107 

Al-Majriti,  109 

Al-Nadim,  102 

Al-Nirizi,  48 

Al-Sagani,  105,  106 

Al-Zarkali,  132 

Amasis,  King,  15 

Amicable  numbers,  56,  104,  109,  239 

Ampere,  A.  M.,  281,  386,  425 

Amsler,  J.,  486 

Amyclas,  29 

Analemma,  48 

Analysis,  method  of,  26,  27,  29.  Modern, 
367-4 n 

Analysis  situs,  211,  285,  323,  324 

Analytic  functions,    257,  258,  425-428,  439 

Analytic  geometry,  40,  159,  162,  163,  167, 
173-184,  224,  275,  276,  293-295,  30J-  i2g 

Anaxagoras,  17;  25 

Anaximander,  16 

Anaximenes,  17 

Andoyer,  H.,  483 

Andrade,  J.,  405 

Andrews,  W.  S.,  366 

Angeli,  Stefano  delgi,  175 

Anger,  C.  T.,  449 

Angle,  triseclion  of.    See  Trisection 


487 


4S8 


INDEX 


Angle,  vertical,  i6 

Anharmonic  ratio.    See  Cross-ratio 

Annuities,  171 

Anthonisz,  A.,  73,  143 

Antinomies,  286,  401,  40a,  4OO 

Anti-parallel,  290,  300 

Antiphon,  23;  24,  51 

Apices  of  Boethius,  52,  68,  lOO^  1 14,  116, 
119,  121 

Apollonian  problem,  41,  144, 179,  288 

ApoUonius,  38-43;  i,  30,  31.  33.  5i.  55.  loi, 
104,  106,  109,  131, 142,  166,  174,  175,  181, 
275 

Appell,  P.,  390;  388,  405 

Approximations  to  roots  of  equations.  See 
Numerical  equations 

Aquinas,  Th,  126;  161 

Arabic  notation.  See  Hindu-arabic  nu- 
merals 

Arabs,  99-112 

Arago,  D.  F.  J.,  260,  269,  275,  368,  441,  470 

Arbogaste,  L..  250;  271 

Arbuthnot,  J.,  244 

Archibald,  R.  C,  33,  41,  246,  275,  301,  480 

Archimedes,  34-39;  i,  28,  30,  31,  zz,  41,  42, 
51.  54.  73,  77.  QO,  loi,  104,  106,  128,  131, 
143,  150,  171,  181,  221,  317,  370.  Archi- 
medean postulate,  35,  327.  Archime- 
dean problem,  107.  Arcliimedian  spiral, 
36,  163.  Cattle-problem,  59.  Mensura 
tion  of  the  circle,  54.  Sand  counter,  54, 
78,90 

.\rchytas,  19;  20,  25,  27,  37,  53 

Arenarius,  54,  78,  90 

Argand,  J.  R.,  265;  254,  349.  420 

Aristaeus,  29,  39 

Aristophanes,  17 

Aristotle,  29;  7,  9,  15,  23,  24,  37,  51,  55,  118, 
126,  129,  i6i,  179,  285,  397.  On  dyna- 
mics, 171.    'His  Physics,  23,  29 

Arithmetic,  Arabic,  102-104,  108,  in. 
Babylonian,  4-7.  Chinese,  71-74,  76,  77. 
Egyptian,  9-14.  Greek,  18,  19,  32,  52- 
62.  Hindu,  8s,  90-93.  Japanese,  78,  79. 
Middle  Ages,  114.  Renaissance,  125,  127, 
128.    Roman,  63-68 

Arithmetical  machines,  206.  272,  483,  485, 
486 

Arithmetical  progression,  5,  12,  13,  58,  75, 
140,  150,  185,  235 

Arithmetical  triangle,  76,  183,  187 

Arithmetization,  362,  398,  369,  424 

Armenante,  A.,  314 

Arnauld,  A.,  170 

Arneth,  A.,  97 

Aronhold,  S.  H.,  346;  348 

"Arrow,"  23 


Aryabhata,  85;  86,  87,  89,  92,  94-96 

Arzela,  C,  377 

Aschieri,  F.,  289,  307,  308 

Ascoli,  G.,  405 

Astroid,  269 

Astrolabe,  48 

Astronomy,  Arabic,  102,  104,  109.  Bahy 
Ionian,  7-9.  Chinese,  76,  77.  Greek,  16, 
ig,  43,  46-48.  Hindu,  83,  84,  95.  Mod- 
ern, 130,  131,  159,  160,  280,  289,  437,  450, 
451-455 

AsjTTiptotes,  40,  142,  177,  185,  188,  224 

Asymptotic  solutions  of  equations,  391,  392. 
454 

AsjTTiptotic  values,  438 

Alabeddin  Jamshid,  no 

Athelardof  Bath,  118 

Atomic  theory,  126 

Atwood,  G.,  155 

Aubrey,  151 

Augustine,  St.,  67 

Ausdehnungslehre,  Z5^,  337 

Axioms,  Geometrical,  11,  26,  31,  32,  48, 
108,  184,  302,  303,  30s,  308.  Algebraical, 
409 

Babbage,  C,  485;  272,  405,  486 

Babbage,  H.  P.,  486 

Babylonians,  2,  4-8,  17 

Bacharach,  M.,  472 

Bachet  de  Meziriac,  167;  168,  170,  254 

Bachmann,  P.,  444;  436 

Biicklund,  A.  V.,  321,  325 

Bacon,  R.,  126 

Baer,  K.,  470 

Bagnera,  G.,  360 

Baillet,  J.,  14 

Baire,  R.,  401,  402 

Baker,  H.  F.,  317,  319,  343.  Quoted,  282, 
283,  316 

Baker,  Th.,  107,  203 

Bakhshali  arithmetic,  84,  85,  89,  91,  92 

Ball,  R.  S.,  455;  308 

Ball,  W.  W.  R  ,  204 

Ballistic  curve,  266 

Baltzer,  H.  R.,  321;  304,  341 

Banachievitz,  T.,  453 

Bang,  A.  S.,  300 

Barbier,  E.,  341,  379 

Bar-le-Duc,  Eward.  See  Eward  de  Bar-Io 
Due 

Barnard,  F.  P.,  122 

Barr,  A.,  301 

Barrow,  I.,  188-190;  158,  163,  192,  207,  212 

Bartels,  J.  M.,  434 

Basset,  A.  B.,  319,  320,  457,  461,  460 

Bateman,  H.,  319 

BattagUni,  G.,  354;  308 


INDEX 


4S9 


Bauer,  G.,  470;  312,  365 

Bauer,  G.  N.,  446 

Bauer,  M.,  34S 

Baumgart,  O.,  239,  436 

Bauschinger,  J.,  483 

Bayes,  Th.,  230,  263.  Bayes's  theorem,  377, 

378 
Bayle,  P.,  182 
Beaumont,  E.  de.,  266,  267 
Beaune,  F.  de.,   180;   174,  176,  209,  210; 

B.'s  problem,  2og 
Becker,  G.  F.,  484 
Becker,  K.,  3S1 
Bede,  tlie  Vxaeral)lc,  114;  120 
Beer,  A.,  469 
Beetle,  R.  D.,  395 
Beha-Eddin,  106,  108,  no 
Bellavitis,  G.,  224,  292,  297,  306,  332,  337, 

366 
Beltrami,  E.,  307;  306,  321,  346,  470 
Beman,  W.  W.,  177,  265,  291 
Bendixson,  I.  0.,  426 
Bennett,  G.  T.,  301 
Bentley,  226 
Bergman,  H.,  368 
Berkeley,    G.,    219;    228;      Analyst,    218; 

Lemma,  219 
Bemelinus,  116;  117 
Bernoulli,  Daniel  (born,  1700)  220;  222,  227, 

240,  242,  251,  252,  365,  378,  382,  383,  448, 

464,  465,  477 
Bernoulli,  Jakob  (James,  born  1654)   220, 

221;  Si,  171,  173,  210,  211,  213,  220,  224, 

234.  238;  331.  380,  465;  Numbers  of  B., 

221,   238;   B.'s  theorem,   222,   263,  377; 

Law  of  large  numbers,  222,  380 
Bernoulli,  Jakob  (James,  bom  1758)  220;  223 
Bernoulli,  Johann  (John,  born  1667)   220, 

222;  57,  210,  211,  213,  216-218,  220,  221, 

224,  227,  232,  234-238,  242,  251 
Bernoulli,  Johann  (John,  born  1710)   220; 

223 
Bernoulli,  Johann  Qohn,  born  1744)  220; 

223 
Bernoulli,  Nicolaus  (born  1687)  220;  223 
Bernoulli,  Nicolaus  (born,  1695)  220;  222, 

238 
BemouUis,  genealogical  Table  of,  220 
Bernstein,  B.  A.,  409 
Bernstein,  F.,  401,  443,  445 
Berry,  A.,  344 
BerthoUet,  C.  L.,  270 
Bertillon,  J.,  380 
Bertini,  E.,  307 
Bertrand,  J.  L.  P.,  379;  34°,  37i,  374,  378, 

^82,  38s,  416,  438,  456,  457,  477 
B.rlrand's  jjostulate,  438 


Besscl,  F.  W.,  448;  304.  311,  377.  382,  452 

Bessy,  Frenicle  de,  169,  170 

Bettazzi,  R.,  409 

Betti,  E.,  346;  307,  35-',  358,  417 

Bevan,  B.,  298 

Beyer,  J.  H.,  ia8 

Bezout,  E.,  249;  23s,  253,  259,  361 

Bezoutiant,  249 

Bhaskara,  85;  86-88,  92,  93,  141,  142 

Bumchi,  L.,  321,  325 

Bieberbach,  360 

Bienaymg,  J.,  382 

Bigelow,  F.  H.,  464 

Billingsley,  H.,  130 

Billy,  Jacobo  de.    See  Jacobo  de  Billy 

Binet,  J.  P.  M.,  340;  465 

Bing,  J.,  378 

Binomial  coefficients,  76,  140 

Binomial  theorem,  178,  186,  187,  192,  205, 

212,  213,  221,  222,  238,  374,  411 
Biot,  J.  B.,  27s;  216,  262,  471,  473 
Biquatemions,  307 
Birational  transformations,   295,  314,  316, 

317,  319 
Birch,  S.,  9 

Birkhoff,  G.  D.,  324,  391.  392,  394,  396,  453 
Bjerknes,  C.  A.,  462;  412,  421 
Bjornbo,  A.  A.,  45 
Blasckke,  E.,  381 
Blaschke,  W.,  370 
Bledsoe,  A.  T.,  173 
Blichfeldt,  H.  F.,  360.    Quoted,  360 
Bliss,  G.  A.,  372,  406,  431,  433 
Blumberg,  H.,  348 
Bobillier,  E.,  310 
Bocher,  M.,  394;  2,  363,  387,  39i,  393,  448, 

465.    Quoted,  284,  286 
Bochert,  A.,  357,  359 
Bode,  J.  E.,  384 
Boethius,  67;  52,  53,  68,  113-116,  118,  119, 

127 
Boggio,  T.,  469 
Bohlin,  K.,  454 
Bohniceck,  S.,  443 
Bois-Reymond,  P.  du,  377;  326,  37i,  374" 

376.  393,  400>  425 
Boll6e,  L.,  48s 

Boltzmann,  L.,  326;  176,  474,  477-479 
Bolyai,  J.,  304;  278,  303,  305-307,  44^ 
Bolyai,  \V.,  303;  278,  302,  304,  435 
Bolza,  O.,  201,  319,  372,  413-    Quoted,  371 
Bolzano,  B.,  367;  258 
Bombelli,  R.,  135;  i37.  141.  i47 
Bompiani,  E.,  322 
Boncompagni,  B.,  178 
Bond,  H.,  189 
Bonnet,  O.,  321;  374,  385 


4QO 


INDEX 


Bonola,  R.,  48,  108,  184,  307 

Boole,  G.,  407;  278,  281,  285.  342,  383,  384, 

386,391,408 
Booth,  J.,  312 
Bopp,  K.,  181 
Borchardt,  C.  W.,  418;  425 
Borda,  J.  C,  266 
Borel,  E.,  375,  386,  401.  402   404   427,  481. 

Quoted,  389 
Borelli,  G.  A.,  184 
Bortkewich,  L.  v.    See  Bortkievicz 
Bortkievicz,  v.,  379,  381 
Bortolotti,  E.,  349,  350 
Bosnians,  H.,  77 
Bouguer  P.,  157,  273 
Boundary-value  problems,   270,    2S4,   391, 

396.  430 
Bouniakovski,  V.  J.,  436 
Bouquet,  J.  C,  3S8;  241,  383,  387,  418,  420 
Bouquet  de  la  Grye,  484 
Bour,  E.,  384 

Boussinesq,  J.,  462,  469,  471 
Boutroux,  E.,  410 
Boutroux,  P.,  429 
Bouvelles,  C,  162 
Bowditch,  N.,  262;  338 
Bowley,  A.  L.,  381 
Boys,  v.,  486 
Brachistochrone,  234 

Bradwardine,  T.,  127;  116,  128,  132,  161 
Brahmagupta,  85;  71,  86,  87,  02,  94,  97,  99 
Braikenridge,  W.,  228 
Brancker,  T.,  140,  169 
Braunmiihl,  A.  v.  48;  137,  235,  481 
Bredon,  S.,  128 
Bremiker,  C,  484 
Brennan,  L.,  458 
Bret,  J.  J.,  269 

Bretschneider,  C.  A.,  9;  88,  336 
Brewster,  D.,  191,  193,  201,  461 
Brianchon,  C.  J.,  166,  275,  287,   288,   298 
Briggs,  H.,  150-152;  155,  187,  343 
Brill,  A.,  293,  313,  316,  328,  419,  430.431 

480 
Brill  L.  309,  328 
Bring,  E.  S.,  349 
Brioschi,  F..  345-347;  279.  307,  340,  34 1, 

347.  348,  361,  370,  388,  413,  417,  456 
Briot,  C,  388;  241,  383,  387,  418,  420 
Brisson.  M.  J.,  265 
Brocard,    H.,    298;    299;    B.    points,    299; 

B.  angles,  299;  B.  circle,  299,  300 
Broch,  O.  J.,  414 
Brod6n,  T.,  433 

Brouncker,  W.,  156;  169,  187,  188,    ^8 
Brouwer,  L.  E.  J.,  403,  433 
Brown,  C.,  329 


Brown,  E.  W.,  450,  451,  453,  454.    Quoted, 

450 
Brownlee,  J.  W.,  383 
Brunei,  H.  M.,  301 
Bruno,  Fak  de,  345 
Bruns,  H.,  452,  454 
Brussels  academy  of  sciences,  168 
Bryant,  W.  W.,  451 
Bryson  of  Heraclea,  23;  24 
Bubnow,  N.,  98 
Buchanan,  D.,  453 
Buchheim,  A.,  455;  308 
Buckle,  H.  T.,  190 
Buckley,  W.,  147;  183 
Budan,  F.  D.,  269,  271 
Buffon,  Count  de,  243,  244,  263,  378,  379 
Buhler,  88 
Bungus,  P.,  144 
Burckhardt,  J.  C.  H.,  440 
Burali-Forti,  C,  2S9,  322,  335,  401,  402,  408 
Burgess,  E.,  85 

Burgi,  J.,  152;  i3t,  148,  154,  178 
Biirja,  A.,  155,  258 
Burkhardt,  H.,  280;  350,  252,  318.    Quoted, 

465,  468 
Burkhardt,  J.  P.,  262 
Burmann,  H.,  272 
Burmester,  L.,  297 
Burns,  J.  E.,  352 
Burnside,  W.,  357,  358,  359,  360 
Burrau,  484 
Burroughs,  485 
Busche,  E.,  436 
Buteo,  J.,  143,  156 
Butterworth,  J.,  298 
Biitzberger,  F.,  292 
Buy's-Ballot,  C.  H.  D.,  478 
Byerly,  W.  E.,  470 
Cajori,  F.,  3,  24, 127,  156,  174,  182, 190,  202, 

224,  248,  271,  330,  344,  447 
Calandri,  Ph.,  128 
Calculate,  origin  of  word,  64 
Calculating  engine.    See  Arithmetical  ma- 
chine. 
Calculus,  See  Differential  C,  Integral  C. 
Calculus  integralis,  the  name,  221 
Calculus  of  functions,  405 
Calculus  of  residues,  420 
Calculus  of  variations,  232,  234,  251,  255, 

267,    281,   291,  394,  313,  367,  369-372, 

404,  405,  430,  431,  437 
Calendars,  8,  66,  70,  76,  78,  114,  122,  132, 

144 
Callet,  F.,  266 
Callisthenes,  7 
Cambuston,  H.,  332 
Campano,  G.,  120;  142 


INDEX 


49  r 


Campbell,  G.,  202 

Cantor,  G.,  307-404;  24,  67,  172,  285,  325. 
367,  400,  404,  409,  426,  432,  440,  447- 
Quoted,  447 

Cantor,  M.,  6;  10,  13,  14,  42,  63,  87,  gi,  06, 
loi,  105,  106,  no,  114,  115,  117,  119,  123, 
140,  211,  ?47,  24g,  269,  384 

Capella,  M.,  113 

Capelli,  A.,  365 

Capillarity,  264 

Capiorali,  PL,  314;  319 

Caqud,  J.,  387 

Cardan,  H.,  134-136;  137-141,  145,  147, 
170,  179,  181,  182,  184,  1S5 

Carette,  A.  M.,  268 

Carlini,  F.,  452 

Carll,  L.  B.,  370 

CannichacI,  K.  D.,  391,  ;:96.    Quoted,  301 

Carnot,  L.  N.  jM.,  27O;  46,  219,  221,  287, 
302,  349. 

Carnot,  S.,  475,  476;  473 

Carra  de  Vaux,  98 

Carslaw,  II.  S.,  48,  108,  465 

Cartan,  E.  J.,  339,  358 

Carvallo,  E.,  335,  365 

Casey,  J..  314:324 

Casorati,  F.,  346;  307,  383 

Cassini,  D.,  244;  190,  222,  245,  451 

Cassini,  J.,  244 

Cassini's  oval,  221,  245 

Cassiodorius,  68,  113 

Castellano,  F.,  409 

Castelnuovo,  G.,  316;  317,  318,  431 

Casting  out  nines,  59,  91,  103 

Catalan,  E.  C,  341;  o30,  383.  47° 

Cataldi,  P.  A..  147,  184,  254 

Catenary,  183,  217 

Cattle  problem,  59,  60 

Cauchy,  A.  L.,  368-370;  227,  232,  238,  249, 
253,  258,  265,  287,  337,  340,  341,  349, 
354.  361-363,  367,  373.  374,  376,  383- 
389,  395.  396,  405,  412,  416,  417,  419.  420, 
426,  428,  430,  431,  438,  440,  446,  452,  461, 
465-468,  471,  473.  Cauchy's  theorem  on 
groups,  352.  Cours  d'analyse,  369,  373. 
Tests  of  convergence  of  series,  373 

Caustic  curves,  222,  225 

Cavalieri,  B.,  161,  162;  79,  159,  162,  165, 
175.  177,  190,  191.  207 

Cayley,  A.,  342-34S;  240,  278,  290,  291,  293, 
295-297,  302,  308,  310,  j'.i2-3i4,  316-320, 
323,  332-335.  338-340,  3SI-353,  361,  383, 
415,  417,  418,  432,  456.  Cayley  line,  291, 
Quoted,  280.  Sixth  memoir  on  quantics, 
307,  308 

Celestial  element  method,  75-80 

Center  of  gravity,  289 


Center  of  oscillaliou,  183,  227 

Center  of  similitude  of  circles,  275 

Centrifugal  foric,  172,  183,  200,  24A 

Ccsaru,  E.,  324.  375.  379 

Ceva,  <;.,  277 

Chamberliii,  T.  C,  450 

Chamixjllion,  11 

Chat\ce.    See  Probability 

Chandler,  S.  C,  241 

Chang  Ch'iu-chien,  73 

Chang  Chun-Ch'ing,  88 

Chang  T'sang,  71;  97 

Characteristic  triangle,  189,  207 

Chapman,  C.  H.,  340 

Charlier,  C,  380,  452,  453 

Charpit,  P.,  255 

Chasles,  M.,  292-294;  a,  39-41.  43,   162. 

174,  246,  276,  287,  295,  297,  308,  310,  312, 

314,  319,  418,  455.  472,  473 
Chauvenet,  W.,  383;  455 
Chebichev,  P.  L.,  380;  301,  344,  438 
Ch'  eng  Tai-wei,  76 
Chernac,  L.,  439 
Chevalier  de  Mere,  170 
Cheyne,  G.,  194 
Child,  J.  M.,  189 
Ch'  in  Chiu-shao,  74;  75 
Chinese,  71-77;  17,  84.     Solution  of  equa- 
tions, 74,  75,  271.    Magic  squares,  76,  77 
Ching  Ch'  ou-ch'  ang,  71 
Chittenden,  E.  W.,  395 
Chiu-cliang,  71 
Chladni,  464 
Choquet,  C,  363 
Chou-pei,  71 
Chree,  C,  461,  469 
Christina,  Queen,  179 
ChristolTel,  E.  B.,  314;  346,  356,  431,  470 
Chrystal,  G.,  378;  379 
Chuproff,  A.  A.,  379 
Chuquet,  N.,  125,  178 
Chu  Shih-Chieh,  75;  76 
Cipher,  origin  of  term,  121 
Circle,  20,  22,  23,  25,  42,  104,  143,  297-300, 

370.     Nine-point  Circle,    298.     Division 

of,  107,  350,  414,  435.  436 
Circle-squarers,  1,  331-    See  Quadrature  of 

the  circle. 
Circular  points  at  infinity,  282 
Circumference,  297 
Cissoid,  42,  51,  182 
Clairaut,  .■\.  C,  244;  227,  229,  239,  242,  245, 

252,  302,  457.    His  differential  equat.,  245 
Clapeyron,  B.  P.  E.,  467,  475 
Clarke.  A.  R.,  379 
Classes,  theory  of,  410 
Clausen,  T.,  330;  22 


492 


INDEX 


Clauslus,  R.,  476;  468,  474,  477-479 

Clavius,  C,  144;  47,  143,  158,  181,  184 

Clayton,  H.  H.,  464 

Clebsch,  R.  F.  A.,  313,  314;  291,  296,  311, 
316,  318,  319,  337,  345-348,  369,  371,  384. 
422,  431,  457,  462,  468,  469 

Clifford,  W.  K.,  307;  278,  303,  308,  317,  333, 
335,  340,  348,  422,  423,  455,  472 

Cockle,  J.,  321 

Colburn,  Zerah,  169 

Colding,  L.  A.,  475 

Cole,  F.  N.,  347,  354,  358 

Colebrooke,  H.  T.,  85 

CoUa,  133,  135 

Collins,  J.,  192,  193,  203,  209,  212-216 

Colson,  J.,  193 

Combescure,  E.,  315 

Combinations,  theory  of,  128,  170,  183,  321 

Combinatorial  school,  231,  232 

Commandinus,  F.,  141;  175,  184 

Commercium  epistoHcum  (Collins')  194, 
215,  216 

Commercium  epistohcum  (Wallis')    168. 

Compensation  of  errors,  219 

Complex  variables,  420,  422 

Comte,  A.,  285 

Conchoid,  42,  51,  202 

Condorcet,  N.  C.  de,  244;  252,  266,  380 

Cone,  27,  33,  39,  46,  79,  141,  319 

Congresses,  international.  See  Interna- 
tional c. 

Conies,  27,  29,  33,  36,  38-41,  so,  51,  88,  141, 
142,  160,  165-167,  181,  184,  246;  Con- 
jugate diameters,  41;  Foci,  40,  41,  160; 
Generation  of,  180,  228;  Names  ellipse, 
parabola,  hyperbola,  39;  Name  latus 
rectum,  40 

Conoid,  36 

Conon,  34;  36 

Conservation  of  areas,  240 

Conservation  of  vis  viva  or  energy,  183 

Constructions,  2,  21,  22,  27,  47,  84,  86, 
106,  297,  300,  310,  336,  436;  By  com- 
passes only,  268;  By  insertion,  36;  By 
ruler  and  compasses,  124,  174,  177,  202, 
292,  350,  436,  446;  By  ruler  and  fixed 
circle,  291;  By  single  opening  of  com- 
passes, 106;  Of  maps,  295;  Of  regular 
polygons,  47,  128 

Contact-transformation,  354,  355 

Conti,  A.  S.,  216 

Continued  fractions,  iii,  147,  188,  246,  258, 
375 

Continuity,  22,  24,  29,  94,  160,  184,  185, 
211,  218,  282,  283,  287,  318,  326,  367,  391, 
399,  419-421,  426 

Continuum,    24,    35,    126,    285,    397,    398, 


400;  Well-ordered,  401;  Not  denumerable, 

402 
Convergence  of  series.    See  Series 
Convergence  of  aggregates,  398 
Convergent  series,  use  of  term,  228,  237 
Coolidge,  J.  L.,  300,  308 
Coordinates,  40,  42,  174,  175,  184,  211,  235, 

2S9,  294,  310,  314,  321,  324,  482;  Elliptic, 

456;  Generalized,  255;  Homogeneous,  297; 

Intrinsic,  324;  Oblique,  174;  Peutaspher- 

ical,   315;   Polar,    221;   Tangential,   310; 

Trilinear,  310;  Movable  axes,  321 
Copernicus,  N.,  46,  130,  131 
Cosserat,  E.,  315,  325 
Cotes,  R.,  226;  199,  236,  382;  Theorem  of, 

228 
Counters,  122 

Counting  board,  75,  90,  91,  122 
Courant,  R.,  433 
Cournot,  281 
Courtivron,  227 
Cousin,  P.,  429 
Cousinery,  B.  E.,  296 
Couturat,  L.,  286 

Covariants,  345,  348,  349,  356,  417,  440 
Cox,  H.,  308 
Craig,  C.  F.,  319 
Craig,  J.,  171,  210 
Craig,  T.,  418;  308,  391,  461 
Cramer,   G.,    241;   175,   204,   223,  320;  C. 

paradox,  228 
Crelle,  A.,  411;  289,  290,  298,  299,  418 
Cremona,  L.,  295,  296;  278,  2S7,  291,  307, 

314,  318,  319,  346.;  C.  transformation, 

295 
Crew,  H.,  172 

Crofton,  M.  W.,  379;  380,  382 
Crone,  C,  320 

Cross-ratio,  166,  289,  293,  294,  297,  308 
Crozet,  C,  276 
Ctesibius,  43 

Cube,  duplication  of.  See  DupHcation 
Cube  root,  71,  74,  123 
Cubic  curves,  204,  228,  229,  244,  249,  295, 

320 
Cubic  equations,  74,  107,  no,  in,  124, 133- 

138,  140,  142,  177,  247,  350 
Culmann,  K.,  296;  294,  297 
Cuneiform  writing,  4,  7,  8 
Cunningham,  A.  J.  C,  446 
Curtze,  M.,  296;  73,  123,  170 
Curvature,  theory  of,  275,  296,  320,  321 
Curves,  163,  202,  204,  206,  207,  209,  224, 

226,  228,  235,  244,  250,  275.  .'95,  318-320, 

321;  Algebraic,  302,  419;  Ballistic,  266; 

Catenary,  183,  217;  Caustic  curves,  222, 

225;  Class  of  curves,  288;  Conchoid,  42, 


INDEX 


493 


SI,  202;  Courhe  du  diable,  ,ui;  Cubics, 
204,  228,  229,  244,  249,  29s,  320;  Defi- 
ciency, 313,  317;  Definition  of  curve,  325, 
326;  Elastic  cun'e,  221,  291;  Fourth  or- 
der, 310,  313;  Fourteenth  order,  312; 
Genus,  316,  320;  Hippopede,  42;  Logarith- 
mic curve,  156,  183,  236;  Logarithmic 
spiral,  156,  221;  Loxodronaic,  142,  221; 
Multiple  points,  224,  229;  Of  pursuit,  273; 
Of  swiftest  descent,  222;  Order  of,  288; 
Panalgebraic,  320;  Peano  curve,  325; 
Polar  curve,  307;  Prony  curve,  301; 
Quartic,  1S8,  241,  245.  319.  320;  Quintic, 
241;  Rectification  of,  181,  182,  221,  224, 
225;  Second  degree,  290;  Space  curves, 
SO,  295,  322;  Third  degree,  310,  312; 
Third  order,  310,  312;  Three-bar-curve, 
301;  Transcendental,  21,  320;  Twisted, 
321;  Versiera,  250;  Witch  of  Agnesi,  250; 
Without  tangents,  326 

Cusanus,  N.,  143 

Cyclic  method  (Hindu),  95,  96 

Cycloid,  162,  165,  166,  177,  181-184,  188, 
210,  217 

Cyzicenus,  29 

Czuber.  E.,  378,  379,  381,  382 

Dale,  4S4 

D'Alembert,  J.,  241-245;  220,  233,  237,  251, 
252,  257-259,  269,  306,  369,  405,  457,  464, 
480;  D.  principle,  242 

Damascius,  51:32,  loi 

Dandelin,  364;  365 

Darboux,  J.  G.,  315;  301,  310,  314,  319,  321, 
322,  325,  347,  354,  355,  372,  383.  386,  406, 
425.  433.  470;  Quoted,  276,  279,  287-289, 
292 

Darwin,  G.  H.,  449;  450,  453,  462,  469 

Dase,  Z.,  440 

Davies,  T.  S.,  271,  29S 

Da  Vinci,  Leonardo;  See  Vinci,  da 

Davis,  E.  W.,  308 

Davis,  W.  M.,  463 

De  Beaune,  F.,  180;  174,  176 

Decimal  fractions,  5,  119,  147,  148 

Decimal  point,  148 

Decimal  system,  4-6,  11,  72,  88 

Decimal  weights  and  measures,  148,  256; 
Dec.  subdivision  of  degree,  148,  484; 
Centesimal  subdivision  of  degree,  152, 
482-484 

Dedekind,  J.  W.  R.,  397-39Q;  32,  35,  172, 
285,  331,  339,  348,  354,  357,  362,  400,  401, 
407,  421,  431,  442,  445,  470 

Definite  integrals.   See  Integrals 

Degree,  decimal  subdivision  of,  148,  483, 
484;  Centesimal  subdivision  of,  152, 
4S2-4S4;  Sexagesimal,  d  152,  483 


De  Qua.  J.  P.,  224;  175 

Dehn,  M.  W.,  327 

DeLahire,  166;  141,  167,  170,  222,  273,  288 

Delamain,  R.,  158,  159 

Delambre,  J.  B.  J.,  437;  484 

Delaunay,  C.  E.,  449;  369,  370,  451 

Delboeuf,  J.,  302 

Del  Ferro,  S.,  133;  i34 

Delian  problem,  21,  27 

Del  Pezzo,  P.,  307 

Del  Re,  A.,  319 

Demartres,  G.,  315 

Democritus  of  Abdera,  25;  15 

De  Moivre,  A.,  229;  222,  224,  377,  380; 
De  Moivre's  problem,  230 

De  Morgan,  A.,  330;  32,  159,  194,  196,  212, 
215,  233,  250,  271,  273,  278,  323,  331,  369, 
374,  377,  379.  381.  382,  405,  407;  Quoted, 
I.  2,  57.  95,  149.  172.  213,  217,  263,  363; 
Budget  of  Paradoxes,  332 

Demotic  writing,  1 1 

Demoulin,  A.,  315 

Denton,  W.  W.,  322 

Derivatives,  method  of,  258 

Desargues,  G.,  166;  141,  146,  164,  167,  174, 
273,  285,  287;  Theorem  of,  285,  327 

Desboves,  A.,  456 

Descartes,  R.,  173-184;  2,  40,  50,  107,  141, 
146,  156,  162-167,  172,  181,  190,  192,  200, 
205,  207,  209,  239,  240,  273,  276,  310, 
355.  3C'.  363,  401;  Folium  of,  177,  229; 
Ovals,  176;  Rule  of  Signs,  178,  179,  201, 
224,  248 

Descriptive  geometry,  246,  274-276,  296, 
297,  308,  327;  Shades  and  shadows,  297 

De  Sluse.    See  Sluse 

De  Sparre,  459 

Determinants,  80,  211,  249,  254,  264,  266, 
312,  314,  340-342,  362,  370,  434;  Name 
"determinant,"  340;  Skew,  340;  Pfaf- 
fians,  340;  Infinite,  341,  394 

Devanagari  numerals,  100,  loi 

De  Witt,  J.,  180 

Dichotomy,  23 

Dickson,  L.  E.,  318,  339,  348.  357-359.  360, 
442;  Quoted,  443 

Differentiability,  376,  399,  423,  425 

Differential  Calculus,  3,  41,  163,  191,  196, 
201,  208-210,  276,  393,  320,  367-369. 
Controversy  on  invention  of,  212-218; 
Japanese,  79 

Differential  coefficient,  first  use  of  word,  272 

Differential  equations,  164,  195,  196,  208, 
211,  222-225,  227,  238,  239,  243,  24s,  254, 
25s,  263,  264,  282,  324,  332,  367,  371,  372, 
373,  383-391-  394.  396,  405.  417.  432,  45O: 
456;  Hyper-geometric,  282;  Linear,  238, 


494 


INDEX 


263,   391,    303.     See  Partial   differential 

equations,  Singular  solutions.  Differential 

calculus,  Integral  calculus,  Three  bodies 

(problem  of). 
Differential   geometry,   306,   315,   321,   322 
Differential  invariants,  345,  355,  356,  388 
Dingeldey,  F.,  323 
Dini,  37s;  279,  377,  43i 
Dinostratus,  27;  21 
Diodes,  42 
Diodorus,  9,  34 
Diogenes  Laertius,  9,  16 
Dionysodorus,  45 

Diophantine  analysis,  62,  81,  95,  168 
Diophantus,  60-62;  45,  48,  51,  87,  93-95. 

loi,   103,   105,   106,   III,   13s,    167,   168, 

401 
Directrix,  50 
Dirichlet,  P.  G.  L.,  438;  168,  170,  270,  278, 

357,  362,  372,  376,  377,  392,  400,  412,  418, 

4ig,  421,  422,  424,  429,  439,  442,  443,  470; 

D.  principle,  2S4,  392,  422,  428,  429,  430, 

433,  473 
Discriminant  (name)  345 
Distance,  308 

Divergent  series,  228,  237,  238,  242 
Division  of  circle,  107,  350,  414,  435,  436 
Division  of  numbers,  7,  73,  117,  119 
Diwani-numerals,  100 
D'Ocagne,  M.,  482;  483,  485 
Dodd,  E.  L.,  382 
Dodgson,  C.  L.,  302 
Dodson,  J.,  15s 
Donkin,  W.  F.,  456;  465 
Dositheus,  34 
Dostor,  G.  J.,  341 

Double  false  position.    See  False  position 
Dove,  H.  W.,  463 
D'Ovidio,  E.,  308,  341 
Drach,  S.  M.,  366 
Drobisch,  M.  W.,  224 
Dronke,  A.,  311 

Duality,  principle  of,  288,  290,  294,  310 
Dubois-Ayme,  273 
Duffield,  W.  W.,  483 

Duhamel,  J.  M.  C,  383;  363,  369,  416,  467 
Duhem,  P.,  127;  128 
Diihring,  E.,  183 
Duillier.    See  Fatio  de  Duillier 
Dulaurens,  F.,  225 
Dumas,  W.,  348 
Duns  Scotus,  126 
Duodecimals,  63,  64,  117,  119 
Dupin,  C,  275;  296,  320,  379;  D.  theorem, 

275 
Duphcation  of  a  cube,  2,  19,  20  21,  27,  38, 

42,  142,  177,  202,  2J6 


Dupuy,  P.,  351 

Durege,  H.,  311;  418 

Durer,  A.,  141;  170,  145 

Dyck,  W.,  324,  329,  432 

Dyname,  335 

Dynamics,  171,  172,  183,  223.  255,  307,321, 

332,  477.    See  Potential 
Dziobek,  O.,  453,  455 
Earnshaw,  S.,  462 

Earth,  figure  and  size  of,  102,  229,  281 
Earth,  rigidity  of,  240,  469 
Ecole  normalc  founded,  256 
Ecole  polyteclinique  founded,  256 
Eddy,  H.  T.,  296 
Edgeworth,  I'.  Y.,  378,  381 
Edleston,  J.,  212,  216 
Eells,  W.  C,  70 
Egyptians,  9-15,  17-19 
Ehrenfest,  481 

Einstein,  A.,  335,  479,  480,  481 
Eisenhart,  L.  P.,  319,  321;  Quoted,  315 
Eisenlohr,  \.,  9,  10 
Eisenlohr,  F.,  369. 
Eisenstein,  F.  G.,  440;  346,  348,  417,  421^ 

436,  441,  442,  444 
Elastic  curve,  221,  296 
Elasticity,  460,  464-470 
Eliminant,  249 
Elimination,  311,  312,  361 
Elizabeth,  Princess,  179 
Elliott,  E.  B.,  348 
Ellipsoid,  attraction  of,  229,  244,  263,  266, 

267,  273,  293,  437 
Elliptic  functions,  225,  232,  239,  291,  313, 

314,  362,  390,  441,  414-417,  434,  437; 

Addition-theorem,    291;   Double  periodi- 
city, 414 
Elliptic  integrals,   239,   258,   266,  267,  414 
Ellis,  A.  J.,  155 
Ellis,  L.,  152 
Ely,  G.  S.,  444 
Emsh,  A.,  300 
Emmerich,  A.,  300 
Encke,  J.  F.,  452;  364,  377,  382,  437 
Encyclopedie  des  sciences  math.,  280 
Encyklopiidie  d.  math.  Wiss.,  280 
Enestrom,  G.,  128,  140,  148,  158,  173,  174, 

179,   184,   221,   223,    225,   233,  23s,  239, 

439 
Engel,  F.,  184,  354,  355,  356 
Enneper,  A.,  416,  417 
Enriques,  F.,  316,  317,  322,  328,  446,  481 
Entropy,  476 

Enumerative  geometry,  292,  293,  295 
P^nvelopes,  theory  of,  211 
Epicycloids,  141,  166,  224 
Epimenides  puzzle,  402 


INDEX 


495 


Epping,  J.,  8 

Equations,  theory  of,  138,  139, 156,  201,  240, 
253.  254,  264,  344,  347,  349-366;  Abelian, 
411;  Cubic,  74,  107,  no.  III,  124, 
133-137.  140.  177.  247;  irreducible  rase, 
I3S.  138,  142,  350;  Every  e.  has  a 
root,  3,  237,  253,  349;  Functional,  39S. 
405;  Indeterminate,  60,  73,  74,  94-96,  106, 
124,  167,  441;  Linear,  13,  75,  95,  103,  211, 
393.  394;  Modular,  352,  416,  417;  Nega- 
tive roots  of,  176;  Of  squared  differences, 
249,  254;  Resultant  of,  249;  I'iule  of  signs, 
178,  179,  201,  224,  248;  Quadratic,  13, 
57;  74,  75.  94,  95,  103,  106,  107,  138; 
Quartic,  61,  107,  75,  i77,  235,  135,  138; 
Resolvents,  138;  Quintic,  253,  332,  349, 
350,  411;  solution  by  elliptic  integrals, 
350.  See  Differential  e.,  Integral  e., 
numerical  e 

EquipoUences,  337 

Eratosthenes,  38;  21,  30,  34,  58.  His 
''sieve,"  58 

Erdmann,  G.,  371 

Ermakoff,  W.,  375 

Errard  de  Bar-le-Duc,  I.,  158 

Escherich,  G.,  v.,  372 

Espy,  J.  P.,  462 

Ether,  theory  of,  460,  461 

Ettinghausen,  A.  v.,  363 

Euclid,  29-34;  18,  22,  25,  29,  39,  46,  53,  59, 
loi,  123,  184,  268,  353,  442 

Euclid's  Elements,  15,  18,  19,  27,  28-35,  44, 
47.  50.  51,  57,  58,  67,  86,  104,  108,  118- 
120,  125,  129,  130,  142,  148,  165,  167,  192, 
205,  226,  302,  303,  307;  Euclid's  Data,  sy, 
Elements  in  China,  77;  Algebra,  61 

Eudemian  summary,  15,  16,  18,  26,  28,  30 

Eudemus,  15,  19,  38,  39,  57 

Eudoxus,  28;  15,  25,  27,  30,  31,  32,  35,  42, 
327 

Euler,  L.,  232-242;  62,  95,  143,  144.  152, 
158;  175,  190,  220,  222,  22s,  225-227,  231, 
232,  245-247,  249,  251-254,  257,  260,  264, 
270,  275,  297,  320,  322,  324,  329,  330, 
353.  36s,  369,  373,  377,  380,  381,  389,  405, 
411,  419,  420,  435,  439,  448,  450,  457,  458, 
464,  465,  477;  'EwXev's  Algebra,  233;Analy- 
sis  situs,  323;  Euler  line,  298;  Infinite 
series,  373;  histitutiones  calculi  diff.,  233, 
239;  Institutiones  calculi  int.,  233,  239; 
Integrating  factors,  239;  Inlroductio  in 
analysin,  227,  233,  241;  Magic  squares, 
170;  Mechanica,  240;  Melhodus  invenicndi 
lineas  curvas,  234;  Method  of  elimina- 
tion, 25;  Number-theory,  168-170,  239; 
Polyedra,  240;  Quadratic  reciprocity,  239; 
Symmetric  functions,   235;    Tlieoria  mo- 


tuum  lunae,  234;  Theoria  motuutn  plane- 

tarum,  234 
Eutocius,  51;  38,  44.  53,  54 
Evans,  G.  C,  395 
Evolutes,  41,  183 
Exchequer,  122 
Exhaustion,  method  and  process  of,  23,  24, 

28,  31,  35,  36,  41,  109,  160,  161 
Exhaustion,  origin  of  name,  181 
Exponential  calculus,  222 
Exponents,    140,    148,    149,   178,   187,   235; 

Fractional,  148,185,  238,  247;  Imaginary, 

225;  Literal,  192;  Negative,  185,  238 
Faber,  G.,  429,  446 
Fabri,  H.,  206 
Fabry,  C.  E.,  375 
Faerber,  C,  366 
Fagnano,  Count  de,  225;  239 
Falk,  M.,  428 
Falling  bodies,  171,  183 
False  position,  12,  13,  91,  93,  103,  137,  366; 

Double,  44,  103,  no,  123 
Fano,  G.,  322,  327,409 
Faraday,  M.,  474,  475 
Farkas,  J.,  405 
Farr,  W.,  383 
Fatio  de  Duillier,  214 
Faye,  H.,  455 
Fechner,  G.  T.,  381 
Fekete,  M.,  362 
Felt,  D.  E.,  48s 
Fenn,  J.,  302 
Fermat,  P.  de,  163-170;  142,  146,  147,  162, 

174-177,  180-182,  189-191,  239,  250,  276, 

401,  438 
Fermat's  theorem,  169,  239,  254 
Fermat's  last  theorem,  106,  168,  239,  254, 

442,  443 
Ferrari,  L.,  135;  i34,  i39,  253 
Ferrel,  W.,  463;  449.  458.  464 
Ferrero,  A.,  382 
Ferrers,  N.,  470 

Ferro,  S.  del.    See  Del  Ferro,  S. 
Ferroni,  P.,  221 
Feuerbach,  K.  W.,  298 
Fibonacci.    See  Leonardo  of  Pisa 
Fiedler,  W.,  297;  313 
Field,  P.,  320 
Fields,  J.  C,  436 

Fifteen  school  girls,  problem  of,  323 
Finck,  Th.,  151 

Fine,  II.  B.,  362,  383;  Quoted,  362 
Finger  symbolism,  63,  65,  68,  114 
Finite  differences,  224,  226,  230,  238,  258) 

264,  405,  408,  466 
Fink,  K.,  291 
Fischer,  E.,  376,  396 


496 


INDEX 


Fisher,  A.,  378 

Fiske,  T.  S.,  279 

Fite,  W.  B.,  357 

Fitzgerald,  G.  F.,  471,  479 

Fleck,  A.,  444 

Floquet,  G.,  348 

Floridas,  133,  134 

Flower,  R.,  155 

Fluents,  ig3,  i94.  iQS.  200,  213 

Fluxions,  150,  192-197,  200,  210,  213,  228, 
247;  Controversy  on  invention,  212-218; 
Berkeley's  attack  on,  219,  220;  Compen- 
sation of  errors  in,  219 

Folium  of  Descartes,  177,  229 

Foncenex,  D.  le.,  237;  257 

Fonctionelles,  395,  405 

Fontaine,  A.,  239;  242 

Ford,  W.  B.,  375 

Forsyth,  A.  R.,  279,  345,  356,  384,  385,  386, 
388,  433;  Quoted,  281,  416,  342 

Foster,  S.,  158 

Foucault,  J.  B.  L.,  473 

Fourier,  J.,  269-271;  164,  234,  242,  247,  281, 
363,  364,  365,  400,  413,  418,  419, 438,  473; 
Analyse  des  equations,  269,  284;  Fourier's 
series,  242,  270,  283,  375-377,  393,  396, 
461,  465;  Fourier's  theorem,  269,  284, 
438;  Theorie  analytique  de  la  chaleur,  270, 
271 

Fourth  dimension,  184,  256,  318,  335,  480 

Foville,  A.  de,  380 

Fractions:  Duodecimal,  63,  64,  117,  iig; 
Partial,  211;  Rational,  22;  Sexagesimal, 
5.  54.  483;  Unit-fractions,  12,  14,  44,  71, 
123;  Continued,  iii,  147,  188,  246,  258, 
375;  Roman,  64;  Chinese,  71;  Decimal, 
S,  119,  147,  148;  Fractional  line,  123 

Francesca,  Pier  della,  128 

Franklin,  B.,  170 

Franklin,  Christine  Ladd,  407 

Franklin,  F.,  343,  345,  436,  444 

Frantz,  J.,  448 

Fr^chet,  M.,  372,  393,  394,  404-406 

Fredhohn,  E.  I.,  393;  394,  341,  427,  469 

Frege,  G.,  408;  286,  407,  409 

Frenicle  de  Bessy,  169,  170 

Fresnel,  A.  J.,  470,  471;  183,  275,  311,  314, 
319,  333,  344,  460,  465,  473 

Frfeier,  A.  F.,  274 

Fricke,  R.,  417,  432,  433 

Friedlein,  G.,  64;  65 

Frischauf,  J.,  452 

Frizell,  A.  B.,  402,  403 

Frobenius,  F.  G.,  354;  339,  341,  347,  353, 
357,  360,  362,  387.  390,  427.  431.  443; 
Quoted,  362 

Frost,  A.  H.,  366 


Froude,  W.,  457;  462 

Fuchs,  L.,  387;  385.  388,  390,  432 

Fuchs,  R.,  279,  347 

Fueter,  R.,  445 

Fujita  Sadasuke,  81 

Functional  calculus,  392,  395 

Functionals,  395 

Functions,  127,  211,  234,  238,  258,  270,  284, 
388,  389,  400,  411-433,  445,  446;  Abelian 
f.,  281,  313,  342,  390,  411,  412,  415,  418, 
419,  421,  423,  424;  Algebraic,  f.,  295,  418, 
430,  431;  Analytic,  f.,  257,  258,  425-428, 
439;  Arbitrary,  f.,  242,  251,  252,  258,  270, 
419;  Automorphic  f.,  432;  Bessel  f.,  448; 
Beta  f.,  234;  Calculus  of,  331,  332;  Com- 
plex variables,  420, 422;  Definition  of,  270, 
326,  400,  419;  Fuchsian  f.,  389,  390; 
Gamma  f.,  234,  416;  Hyperbolic  f.,  246, 
424,  484;  Hyperelliptic  f.,  281,  411,  418; 
Modular  f.,  416,  417,  432;  Multiply- 
periodic  f.,  283;  Non-dillerentiable  f.,  326; 
F.  on  point  sets,  403,  404;  Orthogonal  f., 
396;  Potential  f.,  284,  422;  Sigma  f.,  417; 
Symmetric  f.,  293,  361,  414;  Theta,  :}42, 
415,  416,  418.  See  Elliptic  functions 
Trigonometric  f.,  234,  236;  Zeta  f.,  439. 

Funicular  polygons,  296 

Furstenau,  E.,  341,  365 

Furtwangler,  P.  H.,  443,  445 

Fuss,  P.  H.,  157,  237,  249 

Gaba,  M.  G.,  395 

Galbrun,  H.,  391 

Galileo,  171,  172;  37,  80,  130,  146, 159, 161, 
162,  170,  179,  223 

Galloway,  T.,  382 

Galois.  E.,  351;  352,  353,  354,  358,  411,  432, 
445;  G.  resolvent,  411;  G.  group,  318 

Galton,  F.,  381;  380 

Garbieri,  G.,  341 

Gardiner,  235 

Gauss,  K.  F.,  434-439;  3.  6,  62,  146,  169, 
184,  231,  232,  235,  237,  238,  248,  253,  265, 
278,  281,  284,  289,  295,  314,  320,  322,  325, 
336,  340,  342,  348-351,  353,  361,  366, 
369,  371,  373,  382,  430,  438-440,  442,  446- 
448,  452,  459,  460,  469,  472,  475,  484 
Disquisiliones  arilhmeticae,  435-437;  Non- 
euclidean  geometry,  303-306;  Theoria 
molus,  437,  447 

Gay  de  Vernon,  S.  F.,  274 

Gay-Lussac,  275 

Geber,  109 

Gehrke,  J.,  300 

Geiser,  C.  F.,  318 

Gellibrand,  H.,  151-152 

Geminus,  44;  39,  42,  45,  47,  48 

General  Analysis,  392,  394,  395 


INDEX 


497 


Gcnocchi,  A.,  436 

Gentry,  R.,  320 

Geodesies,  234,  267,  372 

Geometrical  progressions,  s,  7,  13,  140,  150, 
IS4,  185,  235 

Geometrographics,  300 

Geometry,  Analytic,  40,  159,  162,  163,  167, 
173  184,  224,  275,  27''.  203-2OS.  309- 
329;  Analytic  geometry,  rivalry  with 
synthetic,  28S;  Analysis  situs,  211,  285, 
323,  324;  Arabic,  104;  Babylonian,  6,  7; 
Chinese,  76;  Descriptive,  246,  274,  175; 
Differential,  306,  315,  321,  322;  Egyptian, 
9-1 1 ;  Enumerativc,  292,  293;  Geometro- 
grapliics,  300;  Greek,  15-52;  Hindu,  83- 
88;  Ideal  elements  in,  288;  Knots,  322; 
Models,  328,  329;  Non-archimedian,  327; 
Non-desargiiesian,  327;  Non-euclidean, 
32,  302-309;  Of  position,  276,  297;  Of 
n  dimensions,  184,  256,  293,  306, 308,  318, 
321,  322,  333,  337.  480;  Projective,  276, 
285,  292-294,  297,  308,  327,  328;  Roman, 
65;  Shades  and  shadows,  297;  Synthetic, 
166,  167,  286-309 

Gerard,  L.,  436 

Gerard  of  Cremona,  132 

Gerbert,  115;  116-120 

Gerdil,  447 

Gergonne,  J.  D.,  288,  289;  166,  287,  290,  310 

Gergonne's  Annales,  273,  288 

Gerhardt,  C.  I.,  208,  210,  212,  214,  215,  217 

Gerling,  C,  347 

Germain,  S.,  442;  464,  465 

Gerson,  Levi  ben,  128 

Gerstner,  F.  J.  v.,  467 

Ghetaldi,  M.,  174 

Gibbs,  J.  W.,  476,  477;  278,  282,  289,  334, 
452,  465;  Gibbs  phenomena,  465 

Gierster,  J.,  432 

Giorgi,  G.  L.  T.  C,  395 

Girard,  A.,  156;  148,  158,  202 

Giudice,  F.,  408 

Glaisher,  J.,  440 

Glaisher,  J.  W.  L.,  153,  iSS,  34i,  381,  382, 
441,  444,  448,  482,  484 

Glazebrook,  R.  T.,  471,  474 

Globular  projection,  167 

Godfrey,  T.,  204 

Goldbach,  C,  236;  249;  Theorem,  249,  439 

Golden  section,  28,  142 

Gonella,  486 

Gopel,  A.,  418 

Gordan,  P.,  346;  313,  345,  347,  348,  361, 
431,  446;  His  theorem,  347 

Gossard,  H.  C,  298 

Gould,  B.  A.,  383;  451 

Gourn6rie,  J.  de  la,  313,  296 


Goursat,  E.  J.  B.,  385;  279.  325.  360,  372, 

386,  413,  428 
Gow,  J.,  10,  22,  53,  55,  59,  60,  6a,  129 
Grace,  J.  H.,  348 
Graffe,  C.  H.,  364;  365,  366 
Grammateus,  140 
Grandi,  G.,  238;  250 
Graphic  statics,  294,  296 
Grassmann,  H.,  289,  407,  408 
Grassmann,  H.  G.,  335-337;  289,  306,  322, 

332,  338,  407,  408,  455 
Graunt,  J.,  171.  380 
Grav^,  D.  A.,  454 
Graves,  J.,  330 
Gravitation,   law   of,    183,    199,   200,   232, 

259,  260,  262 
Gravity,  center  of,  183 
Gray,  P.,  155 

Greatest  common  divisor,  32,  58,  148,  180 
Grebe,  E.  W.,  299;  471 
Green,  G.,  472;  281,  342,  419,  422,  460-462, 

466,  468,  473 
Green,  G.  M.,  322 
Greenhill,  A.  G.,  344,  417,  4S8,  461 
Greenwood,  S.  M.,  383 
Gregory,  D.,  201,  217 
Gregory,  D.  F.,  273;  330 
Gregory,  J.,  143;  is6, 189, 190.  206,  212,  216, 

225,  228,  238 
Griffith,  F.  L.,  10 
GronwaIl,T.H.,46s 
Grossmann,  M.,  480 
Grote,  24 
Grotefend,  4 
Groups,  253,  282,  283,  285,  335.  346,  347, 

349-366,  388,  438,  480;  Continuous,  282, 

306,  339.  355.  417,  357;  Isometric,  325; 

Use  of  word,  351;  Abstract,  352,  353,  360; 

Of  regular  solids,  353;  Primitive,  354,  359; 

Solvable,  357;  Simple,  358,  360;  Linear, 

358,  359 
Grunert,  J.  A.,  320;  248,  336 
Gruson,  J.  P.,  268 
Gua,  dej.  P.,  175.  248 
Gubar-numerals,  68,  100 
Guccia,  G.  B.,  296 
Gudermann,  C,  424;  4i7,  484;  Guderman- 

nian,  424 
Guerry,  A.  M.,  381 

Guichard,  C,  315.  392,  321,  322,  325,  426 
Guimaraes,  R.,  142 

Guldin,  P.,  159;  49,  161;  His  theorem,  159 
Gundelfinger,  S.,  484;  366 
Gunter,  E.,  151 

Giinther,  S.,  i,  m,  115.  204,  250,  341,  365 
Guthri,  F.,  323 
Giitzlaff,  C.  E.,  417 


4q8 


INDEX 


Oyl  Jen,  TI.,  453;  454 

Haan,  D.  B.  He,  372 

Haas,  A.,  321 

Habenicht,  B.,  250 

llachette,  J.  N.  P  ,  276;  275,  2g6 

Hadamard,  J.,  372;  324,  34°,  375,  3Q5.  402, 

427,  43g;  Quoted,  3g2 
Hadley,  J.,  204 
liagen,  G.  H.  L.,  382 
Hahn,  H.,  372,  4cx>,  406,  42g 
Hahn,  Ph.  M.,  485 
Halifax,  J.,  127 
Hallam,  13Q 
Halley,  E.,  156;  38,  41,  171,  igo,  igg,  200, 

203,  2ig,  22g,  251,  343,  380,  381,  448 
Halphen,  G.  H.,  313;  2g3,  321, 322,  345,  355, 

375,  388,  3go,  417 
Ilalsted,  G.  B.,  130,  304,  327,  389,  390,  425; 

Quoted,  446 
Hamburger,  A.,  387 
Hamburger,  M.,  383 
Hamel,  G.,  405 

Hamilton,  W.,  172,  173,  273,  278,  331 
Hamilton,  W.  R.,  332;  255,  280,  314,  322, 

330,  331,  333,  335,  337,  338,  340,  342,  347, 

349,   384,   352,   353,   455,   4S6,   47i,   473, 

477;  Conical  refraction,  332,  471;  Hamil- 

tonian  group,  357 
Hammond,  J.,  345,  34S 
Hancock,  H.,  371,  372 
Hankel,  H.,  423:  10,  25,  52,  57,  62,  gi,  g3- 

96,  102,  105,  IIS,  120,  i2g,  135,  141,  226, 

273,   337,   341,   3^7,  376,   400,   404,   423; 

Principle   of   permanence,   337;   Quoted, 

2go,  367 
Hansen,  P.  A.,  44g;  382,  451,  452,  486 
llanus,  P.  H.,  541 
Hann,  J.,  463,  464 
Hardy,  A.  S.,  265 
Hardy,  C,  164 
Hardy,  G.  H.,  439 
Harkness,  J.,  433 
Harnack,  A.,  404 
Harmonics,  theory  of,  40,  46 
Harmuth,  Th.,  366 
Harpedonaptae,  10,  25 
Harrington,  M.  W.,  455 
Harriot,  T.,  156,   157;   137,  141,  149,  158, 

178,  i7g,  184 
Hart,  A.  S.,  2g8;  2gi,  318 
Hart,  H.,  301 
Hartogs,  F.  M.,  401,  429 
Harzer,  P.,  452 
HaskeU,  M.  W.,  310 
Haskins,  C.  N.,  356 
HaUidakis,  N.,  321 
Hawkes,  H   E.,  ;  jg 


Hayashi,  T.,  78,  82 

Hazlett,  O.  C,  330 

Hearn,  155 

Heat,  theory  of,  270,  3gi,  470-479 

Heath,  R.  S.,  308 

Heath,  T.  L.,  32,  41,  60,  62,  168,  302 

Heaviside,  O.,  334,  474,  475 

Heawood,  P.  J.,  323 

Hebrews,  7,  17 

Hecke,  E.,  433 

Hecker,  J.,  349 

Hedrick,  E.  R.,  328,  372,  430 

Heffter,  L.,  324,  480 

Hegel,  447 

Heiberg,  J.  H.,  34,  35,  44 

Heine,  E.,  377;  397,  398,  400,  470 

Hellinger,  E.,  406 

Helmholtz,  H.,  474-477;  459,  460,  461,  463, 

464,  471,  477 
Henderson,  A.,  317,  318 
Henrici,  0.,  422,  423 
Henry,  J.,  473 
Hensel,  K.  W.  S.,  431,  445 
Herigone,  P.,  205 
Hermann,  J.  M.,  486 
Hermes,  O.,  436 
Hermite,  C,  415,  416;  4,  7,  279,  345,  346, 

348,  350,  375,  388,  391,  412,  413,  418,  420, 

432,  433,  444-446,  465,  470 
Hermotimus,  28 
Hero.    See  Heron 
Herodiauic  signs,  52 
Herodianus,  52 
Herodotus,  quoted,  g,  11 
Heron,  43-45;  42,  54,  61,  66,  84,  86,  loi,  114, 

131 
Heron  the  Younger,  43 
Herschel,  J.  F.  W.,  464;  126,  272,  405 
Hertz,  H.  R.,  281,  474 
Hertzian  waves,  3g3 
Hess,  W.,  458 
Hesse,  L.  0.,  311,  312;  313,  320,  341,  346, 

351,  361,  36g,  384,  452 
Hessel,  L.  0.,  2gi 
"Hessian,"  312,  345 
Hettner,  G.,  425 
Heuraet,  H.  van,  181 
Hexagon,   6,   18,   166,   228,   2go,   2gi,  318, 

327 
Hexagrammum  mysticum.    See  Hexagon 
Heywood,  II.  B.,  383 
Hicks,  W.  M.,  460,  462 
Hieratic  writing,  11 
Hieroglyphics,  11 
Hilbert,  D.,  279,  309,  325,  326,  328,  341, 348, 

372,  391,  393-395,  404,  430,  433,  443,  445. 

446;  Quoted,  430 


INDEX 


400 


(fil.lehrandt,  T.  H.,  406 

Hill,  (;.  F.,  T2I 

Hill,  C.  W.,  450;  394,  341,  4ST,  4S3 

Hill,  J.  E.,  319 

Hill,  J.  M.,  384 

HUl,  Th.,  324 

Hilprecht,  H.  V.,  7 

Hilton,  H.,  3O0 

llindenburg,  C.  F.,  373;  272 

(lindu- Arabic  numerals,   2,  52,  68,  SS-go, 

98,  100,  loi,  107,  120,  121,  128,  147 
llindus,  83-g8;  2,  8;  Geometry,  83-86 
(lipparchus,  43;  5,  45-471  141,  481 
llippasus,  19 
Ilippias  of  Elis,  21 
Hippocrates  of  Chios,  21;  22,  23,  25,  26,  30, 

51,  57,  61 
dippolytos,  59,  91 
Hippopede,  42 
Hirn,  G.  A.,  476 

History  of  Math's,  why  studied,  1-3 
Hobson,   E.  VV.,   22,    144,    268,   446,   470; 

Quoted,  30S 
Hodgkinson,  E.,  467 
Hodograph,  332 
Hoernly,  R.,  85 
Holder,  O.,  357,  358,  427 
Holmboe,  B.,  414;  374,  411 
Holzmann,  W.,  141 
Homography,  293 
Homological  figures,  166 
Hooke,  R.,  199,  200,  295 
Hopital,  G.  F.  A.    See  Hospital 
Hoppe,  R.,  309;  453 
Horn,  J.,  387 
Horner,  J.,  366 
Horner,  W.  G.,  75 

Horner's  method,  72,  74,  75,  271,  365 
Horsburgh,  E.  M.,  329,  482,  485,  486 
Hospital,  G.  F.  A.  1',  224;    177,  213,    217, 

222,  244 
Houel,  J.,  31s;  304,  334,  369,  484 
Hsu,  77 

Hudde,  J.,  180;  178,  181,  193 
Hudson,  R.  W.  H.  T.,  319 
Hugel,  Th,  367 
Hughes,  T.  M.  P,  74 
Humbert,  M.  G,  317 
Hunain  ibn  Ishak,  loi 
Huntington,  E.  V.,  357,  395,  409 
Hunyadi,  E,  341 
Hurwitz,  A.,  376,  423,  426,  432,  444,  445, 

446 
Hurwitz,  W.  A,  395 
Hussey,  W.  J  ,  455 
Hutchinson,  j.  I,  319 
Hutton,  C,  247;  154,  155 


Huxley,  344 

Tluygens,  C.  182    183;  143,  166,   169-171, 

173,   179,   188,   190,   199,   2CX5,   205,  206, 

217,   221,   225,  230,   244,  470;  Cycloidal 

pendulum,  166 
Hyde,  E.  W.,  250,  337 
Hydrodynamics,  223,  240,  242 
Hydrostatics,  37 
Hypatia,  50;  31 
Hyperbola,  185,  188,  206,  247;  Equilateral, 

188.    See  Conies 
Ilyiierbolic  functions,  246,  424,  484 
Hyperelliptic  functions,  281,  411,  418 
Hypergeometric  equation,  282 
Hypergeometric  series,  185,  238,  373,  385, 

386 
Hyperspace.   See  Geometry  of  n  dimensions 
Hypsicles,  58;  5,  32,  42,  43,  loi 
lamblichus,  59;  6,  9,  19,  45,  iii 
Ibbetson,  W.  J,  469 
Ibn  Albanna,  110 
Ibn  Al-Haitam,  104,  107,  109 
Ibn  Junos,  109 
Ideal  numbers,  442,  443 
Ikeda,  80 
Imaginaries,   225,   226,  229,  236,  237,  275, 

279,  284,  288,  292-294,  315,  332,  390,  420, 

434 

Imaginarj'  roots,  123,  135,  156,  179,  202, 
248,  249,  363-365;  Graphic  representa- 
tion of,  184,  237,  264,  265 

Imamura  Chisho,  79 

Impact,  179 

Imshenets  ki,  W.  G.,  385;  386 

Incommensurables,  22,  28,  31,  32,  57,  126; 
Indeterminate  coefficients,  176,  204 

Indeterminate  equations,  60,  73,  74,  94-97, 
106,  124,  167,  441;  Of  second  degree,  62 

Indeterminate  form  '^,  222.  224 

Indian  notation.    See  Hindu-Arabic 

Indicatrix,  275 

Indivisibles,  126,  161,  162,  165,  172,  175, 
184 

Induction  (math'l),  169,  331 

Infinite  products,  186,  187,  417 

Infinite  series,  75,  77,  80,  81,  106,  127,  172, 
181,  187,  188,  192,  193,  196,  206,  212, 
227,  232,  238,  246,  248,  257,  258,  361, 
367,  373,  425,  434,  411;  Convergence  of, 
227,  249,  270,  284,  367,  373-375.  417 

Infinitely  small,  160,  165,  194-198,  207, 
210,  218,  220,  237 

Infinitesimals,  24,  35,  48,  49,  51,  181,  189, 
19s,  ig6,  198,  210,  257,  258,  399 

Infinitesimal  calculus.  See  Differential 
calculus 

Infinity,  23,  24,  66,  126,  160,  166,  177,  184. 


500 


INDEX 


i8s,  219,  237,  241,  243,  257,  2S3,  287, 
307,  402,  446,  447 

Ingold,  L.,  328,  sss 

Ingram,  J.  R.,  292 

Insurance,  171,  223 

Integral  calculus,  3,  7g,  81,  iCi,  209,  210, 
222,  270,  393,  398,  367 

Integral  equations,  392-394,  405,  406,  413 

Integrals,  36,  316,  376,  386,  388,  424; 
Algebraic,  283;  Definite,  i8g,  237,  263, 
369,  385,  414,  416,  421,  466;  Double,  420; 
Elliptic,  239,  258,  266,  267,  414;  Eulerian, 
267,  272;  Hyperelliptic,  413,  415;  Lebes- 
gue,  406,  407;  Multiple,  284,  392;  Pier- 
pont,  406;  Radon,  406;  Riemann,  406,  407 

Integraphs,  486 

Integrations   ante-dating  the  calculus,   189 

Integro-differential  equations,  392,  395, 
40s 

International  commission  on  teacliing,  356 

International  congresses,  280 

Interpolation,  186,  187,  192 

Invariants,  282,  312,  316,  319,  321,  342- 
348,  349,  351,  355,  356,  388,  417,  444; 
Name  345 

Inverse  method  of  tangents,  180,  207-209 

Inversion,  Hindu  method  of,  92 

Inversion  (in  geometry),  292 

Involute,  183 

Involution  of  points,  50,  166 

Ionic  school,  15-17 

Irrationals,  2,  jg,  22,  32,  43,  57,  61,  86,  93, 
94,  103,  133,  140,  330,  396-400,  483; 
First  use  of  word,  68 

Irrational  roots.    See  Roots 

Ishak  ibn  Hunain,  loi 

Isidorus,  51,  113,  121 

Isochronous  curve,  217,  221 

Isomura  Kittoku,  79 

Isoperimetrical  figures,  42,  221,  222,  234, 
251 

Isothermic  surfaces,  314 

Itelson,  G.,  286 

Ivory,  J.,  273;  469;  His  theorem,  273 

Jabir  ibn  Af]ah,  log;  119 

Jacobi,  C.  G.  J.,  414,  415;  266,  278,  289,  290, 
311,  321,  332,  340,  341,  351,  362,  365,  369- 
371,  377,  384,  385,  388,  411-413,  417,  418, 
421,  424,  42s,  429,  436,  437,  441,  448, 
455,  458,  469,  470;  Jacobian,  345;  Theory 
of  ultimate  multiplier,  456 

Jacobi,  K.  F.  A.,  298;  299 

Jacobo  de  Billy,  158 

Jacobs,  J.,  380 

Jahnke,  £.,335 

fahrbuch  iiber  die  Fortschrilte  dcr  Malhe- 
matik,  278 


Janni.  G.,  341 

Japanese,  78-82 

Jastremski,  381 

Jeans,  J.  H.,  450 

Jellett,  J.  H.,  370;  4S7,  468 

Jerard  of  Cremona,  119;  123 

Jerrard,  G.  B.,  349 

Jevons,  W.  S.,  378;  281,  407 

Joachim,  G.,  See  Rhaeticus 

Joachimsthal,  F.,  424 

Jochmann,  E.,  478 

Johanson,  A.  M.,  433 

John  of  Palermo,  124 

John  of  Seville,  118;  119,  147 

Johnson,  W.  W.,  302 

Joly,  C.  J.,  iii 

Jones,  G.  W.,  484 

Jones,  W.,  155,  158,  235 

Jordan,  C.,  318,  325,  326,  347,  348,  353,  354, 
357-360,  379,  390,  428;  Jordan  curve, 
325,  326;  Jordan's  problem,  359 

Jordanus  Nemorarius,  118 

Josephus,  F.,  79 

Josephus  problem,  79,  459 

Joubert  P.,  417 

Joule,  J.  P.,  475:477-479 

Jourdain,  P.  E.  B.,  205,  212,  215,  257,  399, 
400,  401,  402;  Quoted,  407,  408,  409 

Journal  des  Savans  founded,  209 

Journal  of  the  Indian  Malh.  Society,  98 

Journals  (math'l),  82,  98,  209,  273,  278,  288, 
289,  343,  346,  355,  411,  427 

Jupiter's  satellites,  252 

Jiirgensen,  C.,  414 

Jurin,  J.,  219;  369 

Kant,  I.,  261,  449 

Karpinski,  L.  C.,  68,  88,  89,  102,  103,  121 

Karsten,  W.  J.  G.,  237;  265 

Kasner,  E.,  318,  320,  322;  Quoted,  324 

Kastner,  A.  G.,  434,  435;  204,  248 

Kaye,  G.  R.,  84-87,  91,  94,  96-98 

Keill,  J.,  215;  216,  218 

Kelland,  P.,  461;  474 

Kelvin,  Lord  (Sir  William  Thomson),  472, 
473;  271,  272,  279,  292,  323,  329,  334,  422, 
457,  460,  461,  463,  466,  469-471,  475,  476, 
479,  486;  Thomson's  principle,  284,  422, 
428-430,  433,  473;  Tide-calculating  ma- 
chine, 329;  Vortex  atoms,  323 

Kempe,  A.  B.,  2S6,  302,  323,  324,  409 

Kempner,  A.  J.,  444,  446 

Kepler,  J.,  159-161;  131,  145,  146,  148,  154, 
163,  170,  178,  184,  192,  199 

Kepler's  problem,  252 

Ketteler,  C,  471 

Keyser,  C.  J.,  66,  174,  285 

Khayyam,  Omar,  103,  108 


INDEX 


SOI 


Killing,  \V.,  .-^oS,  347    ^5^ 

Kimura,  S.,  335 

Kinckhuysen's  algebra.  103 

Kirchhoff,  G.  R.,  474;  281,  3Tt.  457.  4S8, 

461,  462,  46O,  471 
Kirkman,  T.  P.,  323;  351;  K.  point,  200 
Klein,  F.,  356:  177,  27S,  203,  306-301;,  311, 

314,  318,  319,  328,  34O,  347.  354.  355.  350. 

360,  38S,  390,  391,  417.  418,  431-433,  442, 

457,  458,  480;  Ikosaeder,  347,  359,  417 
Klugel,  G.  S.,  234 
Knapp,  G.  F.,  381 
Kneser,  A.,  327,  37i,  372,  387,  393 
Knoblauch,  J.,  425 
Knots,  theor>'  of,  323 
Knott,  C.  G.,  333 
Kobel,  122 

Koch,  N.  F.  Helge  v.,  326,  341,  439 
Kochansky,  A.  A.,  367 
Koebe,  P.,  433 
Kohn,  G.,  375,  293,  294 
Konig,  J.,  401;  402,  409 
Konigs,  G.  P.  X.,  302,  315;  (  uoted,  294 
Konigsberger,  L.,  413,  417,  4.8,  348 
Kopcke,  H.  A.,  461 
Korkine,  A.  N.,  444;  384 
Korn,  A.,  469 
Komdorfer,  G.  H.  L.,  314 
Kortum,  L.  H.,  292 
Kossak,  H.,  397 
Kotjelnikoff,  A.  P.,  335 
Kotter,  E.,  2S7,  2S8 
Kotter,  F.,  425 
Kotteritzsch,  T.,  341 

Kovalevski,   Madame,  456;  415,  455,  456 
Kowalewski,  G.,  324,  341,  393 
Kramp,  C,  341 
Krause,  K.  C.  F.,  324 
Krause,  M.,  418 
Krazer,  A.,  418 
Kries,  J.  v.,  378,  379,  381 
Kroman,  378 
Kronecker,  L.,  362;  340,  342,  346,  347,  348, 

350.  353.  354.  359,  398,  423,  424,  431,  436. 

444;  Quoted,  362 
Kronig,  A.  K.,  477,  478 
Kuhn,  H.  W.,  360 
Kuhn,  J.,  205 
Kiihnen,  F..  318 
Kummer,  E.  E.,  442-445;  168,  314,  319,  362, 

375.  385.  424,  436,  438;  K.  surface,  314, 

318.319,328,  418 
Kurushima  Gita,  81 
Kustermann,  W.  W.,  376 
Lacroix,  S.  F.,  324;  255,  272,  274,  336,  383, 

419 
Lagny,  T.  F.  de,  203;  143 


f-agrange,  J.,  250-250;  3,  62,  164,  169,  172, 
190,  191,  203,  219,  227,  229-232,  234,  237, 
240-243,  246,  247,  263,  264,  266,  267, 
276,  292,  295,  306,  311,  3M,  320,  336, 
342,  349,  351-353.  356.  35;,  365.  368, 
369,  371,  372,  382-384,  405,  407,  426,  434. 
435,  449.  452,  '55-458.  460,  461,  464-467, 
477,  480;  Calcul  des  Joncdons,  256; 
Generalized  coordinates,  255;  Theorem  on 
groups,  253;  Mecanique  aualyliqtie,  252, 
257;  Method  of  derivatives,  258;  Resolu- 
tion dcs  equations  num.,  227,  253,  256; 
Theorie  drs  fonctions,  256-258 

Laguerre,  E.,  361;  248,  293,  308,  313,  375 

Lahire,  P.  de.,  166;  141,  167,  170,  222,  273, 
288 

Laisant,  C.  A.,  334 

Lalande,  F.  de,  250 

Lalanne,  L.,  481 

Lalesco,  T.,  393 

Lallemand,  E.  A.,  482 

La  Louvere,  .\.,  165 

Lamb,  H.,  26,  455,  461,  462;  Quoted,  284, 
474 

Lambert,  J.  H.,  245-247;  152,  153,  184,  287, 
302,  30s.  314.  407 

Lambert,  P.  A.,  365 

Lame,  G.,  467-470;  297,  388,  469,  470; 
L.  equation,  416;  L.  functions,  467 

Lampe,  E.,  345;  278 

Landau,  E.,  22,  439 

Landen,  J.,  247;  257 

Landry,  F.,  446 

Laplace,  P.  S.,  259-264;  164,  191,  223,  230- 
232,  245,  252,  254,  256,  258,  266,  271,  273, 
302,  336,  368,  369,  374,  377,  379,  380,  382, 
384,  386,  405,  408,  420,  434,  438,  447-449, 
451,  452.  457.  459.  462,  464.  466,  469,  470, 
472,  473.  475;  Great  inequality,  261;  L. 
coefficients,  263;  L.  equation,  264;  L. 
theorem,  264;  Laws  of  Laplace,  261; 
Mcchanique  Celeste,  261-263,  332,  338, 
374;  Systcme  du  monde,  261;  Theorie 
analylique  des  probabilites,  260,  262,  378 

Laqulere,  E.  M.,  366 

Larmor,  J.,  332 

Lasswitz,  K.,  49 

Last  Theorem  of  Fermat,  106,  168,  239, 
254,  442,  443 

Latham,  M.,  48 

Latin  square,  239,  367 

Laurent,  P.  M.  H.,  47°;  42o 

Lauricella,  G.,  469 

Lavoisier,  A.  L.,  256 

Lawrence,  F.  W.  P.,  446 

Laws  of  motion,  171,  179,  199 

Least  action,  240,  255,  284,  429,  477 


INUKv 


Least  resistance  (solid  of),  201,  234,  284 

Least  squares,  263,  266,  267,  273,  382,  434 

Lebesgue,  V.  A.,  405;  341,  369,  401,  402, 
406,  421,  436 

Lebon,  E.,  389 

Legendre,  A.  M.,  266-268;  231,  232,  246, 
247,  256,  263,  293,  303,  306,  349,  351,  370. 
371,  382,  383,  3-S5.  405>  411-413.  417,  435, 
438,  439.  442.  466,  469,  473,  482;  Gcomc- 
trie,  268,  302;  L.  coefficients,  414;  Num- 
ber theory,  170,  239 

Lehmann-Filh6's,  R.,  383,  453 

Lehmer,  D.  N.,  440,  446 

Leibniz,  G.  W.,  205-219;  3,  51,  80,  146,  158, 
161, 165,  169,  173,  17s,  179,  182,  183, 188, 
191,  193,  196-198,  220,  222,  224,  226, 
236-239,  246,  248,  257,  323,  369,  373,  407, 
408,  410,  485;  De  arte  comhinaloria,  205; 
Notation  of  calculus,  207,  208,  210; 
Other  notations,  211,  157;  Controversy 
with  Newton,  212-21S 

Leitzmann,  H.,  340 

Lemniscate,  188,  221,  245 

Lemoine,  E.,  299;  300,  379;  L.  point,  299; 
L.  circle,  300 

Lemonnier,  P.  C,  256 

Leodamas,  28 

Leon,  28;  30 

Leonardo  de  Vinci,  128 

Leonardo  of  Pisa,  1 20^-125;  13,  104,  110, 
I20-I2S,  127,  128,  141 

Leonelli,  G.  Z.,  484;  i5S,  366 

Lerch,  M.,  428 

Leslie,  J.,  218,  145 

Leuschner,  A.  O.,  452 

Le  Vavasseur.    See  Vavasseur,  Le 

Lever,  37 

Leverrier,  U.  J.  J.,  449;  332,  451 

Levi  ben  Gerson,  128 

Levi-Civita,  T.,  ni,  356,  433.  453.  454 

Levy,  M.,  469;  296 

Lewis,  C.  I.,  408 

Lewis,  G.  N.,  335,  481 

Lewis,  T.  C,  460 

Lexis,  W.,  381;  380;  Dispersion  theory,  381 

Lezius,  J.,  98 

L'Hospital.    See  Hospital 

Li  Ch'  unfeng,  71 

Lie,  S.,  354;  27Q,  306,  307.  3ig,  320.  321, 
346.  347,  354,  355,  356.  357,  384,  388,  414; 
Quoted,  355 

Light,  corpuscular  theory  of,  204;  Wave 
theory  of,  183 

Ligowski,  W.,  484 

Liguine,  V.,  302 

Lilavati,  85,  87, 92 

iJlienthal,  R.  v.,  321 


Limits,  24,  182,  184,  198,  220,  243,  257,  283 
367,  369,  373,  396,  398,  399,  421.  446 

Lindeberg,  J.  W.,  372 

Lindelof,  L.  L.,  370 

Lindemann,  F.,  308,  362,  419,  446 

Linear  transformntions,  282,  289,  340,  342 

Ling,  G.  H.,  357,  3S8 

Linkages,  300-302,  344 

Linteria,  221 

Liouville,  J.,  440;  292,  321,  340,  351,  352, 
364,  3S8,  393,  413,  418,  420,  436,  441, 
446,  456,  473 

Lipkin,  301 

Lipschitz,  R.,  308;  376,  431,  449,  461 

Listing,  J.  B.,  323;  359 

Little,  C.  N.,  323 

Littlewood,  J.  E.,  439 

Lituus,  226 

Liu  Hui,  71,  73 

Lloyd,  H.,  471 

Lobachevski,  N.  I.,  303;  278,  304,  306,  307, 
425 

Local  probability,  244 

Local  value,  principle  of,  5,  69,  78,  88,  94, 
100,  102 

Loewy,  A.,  360 

Logarithmic  curve,  156,  183,  236 

Logarithmic  series,  188 

Logarithmic  spiral,  156,  221 

Logarithms,  140,  149-156,  189,  235;  Com- 
mon, 151;  Computation  of,  153-156,  18S; 
radix  method,  153,  155;  Characteristic, 
152;  L.  of  cross-ratio,  293;  L.  of  imagina- 
ries;  225,  235-237,  243,  330,  "Gaussian," 
484,  Natural  1,  150,  152,  153;  247;  Man 
tissa,  152;  In  China,  77;  Logarithmii 
curve,  156,  183,  236;  Logarithmic  spiral. 
156,  221;  Logarithmic  tables,  482,  483 

Logic,  22,  205,  246 

Lommel,  E.,  449,  471 

London,  F.,  292 

Long,  J.,  15s 

Longomontanus,  C,  169 

Loomis,  E.,  463 

Lorentz,  H.  A.,  479,  481 

Lorenz,  J.  F.,  302 

Lorenz,  L.,  471 

Loria,  G.,  7,  22,  42,  156,  176,  177,  183,  245. 
250,  27s,  293,  301,  308,  320 

Lottner,  E.,  458 

Lotze,  R.  H.,  309 

Loud,  F.  H.,  294 

Love,  A.  E.  H.,  470 

Lovett,  E.  O.;  Quoted,  452,  453 

Loxodromic  curve,  142 

Loxodromic  spiral,  221 

Lucas,  E.,  446;  366 


INDKX 


503 


Lucas  (le  Burpo,  125,  187 

Lucretius,  T.,  66 

Lucllam,  \V.,  302 

Ludolph  Van  Ceulcn.    See  Van  Ceulen 

Lunar  theory.    See  Moon 

Lune,  squaring  of,  22,  57 

Lupton,  S.,  IS5 

Liiroth,  J.,  422;  377 

Mac  Cullagh,  J.,  312;  461,  471 

Mac  Coll,  II.,  408;  37g,  407 

Macdonald,  W.  R.,  150 

Macfarlane,  A.,  335;  o^i,  334.  407.  44i 

Mach  E.,  37;  219 

Machin,  J.,  206;  227 

Mackay,  J.  S.,  299;  M.  circle,  300 

Maclaurin,  C,  228,  229;  177,  202,  220,  226, 

244,   267,   273,    277,   293,   369,  377;    M 

theorem,  226,  228,  365 
Mac  Mahon,  P.  A.,  240,  343,  345,  348,  366, 

436 
Mac  Millan,  W.  D.,  433, 453 
Magic  circles,  76,  77,  79,  80 
Magic  cubes,  79,  81 
Magic  squares,  76,  77,  79,  80,  92,  93,  104 

128,  141,  14s,  167,  366,367 
Magic  wheels,  79 
Magnus,  H.  G.,  459 
Magnus,  L.  I.,  295 
Mahavira,  85;  86,  88,  96,  97 
Maillet,  E.,  354;  357,  359,  366,  442,  444, 

446 
Main,  R.,  455 
Mainardi,  G.,  370 
M'  Laren,  J.,  366 
Malebranche,  N.,  222 
Malfatti,  G.  F.,  291:349 
Malfatti's  problem,  81,  291,  314 
Mangoldt,  H.  von,  314,  439 
Mannheim,  A.,  324 
Manning,  Th.,  155 
Manning,  W.  A.,  360 
Mannoury,  G.,  403 
Mansion,  P.,  384 
Map  coloring,  323,  324 
Map  construction,  48,  167,  295,  314 
Marchi,  L.  de,  464 
Marcolongo,  R.,  335,  469 
Margetts,  481 
Margules,  M.,  464 
Marie,  .^bbe,  266;  252 
Marie,  M.,  294;  43,  162 
Mascheroni,  L.,  268;  47,  269 
Maschke,  H.,  359;  318,  m,  347,  35^ 
Masiama  al-MajrIti,  104 
Mason,  M.,  372,  391 
Massau,  J.,  482 
Mathematical  induction,  142 


MathemMtical  periodicals.    5ee  Journals 

Matlicmatical  physics,  392-394 

Mathematical  seminar,  424 

Mathematical  societies.    See  Societies 

Mathematical  Tables,  482-484 

Mathematics,  definition  of,  285,  286 

Mathews,  G.  B.,  395,  414,  445 

Mathieu,  E.,  353;  417.  358,  469,  475 

Mathieu,  P.,  470 

Matrices,  335.  337,  338,  339,  34°,  344,  4o8 

Matthiessen,  L.,  107,  no,  253 

Maudith,  J.,  128,  132 

Maupcrtius,  P.,  244;  240,  477 

Maurolycus,  141;  142,  145 

Ma.xima  and  minima,  32,  40,  81,  142,  160, 

163,   164,   180,   IQ3,   ig6,   210,  229,   267, 

291,  394,  370,  376,  384 
Maxwell,  C.,  474-479;  271,  270,  281,  296, 

322,  iii,  334,  337,  451,  458,  460,  461,  468, 

471,  472,  477,  479,  486 
Maya,  69,  70 
Mayer,  .\.,  425;  355 
Mayer,  M.,  363 
Mayer,  R.,  475;  449 
McClintock,  E.,  365;  360 
McCoU,  IL    Sec  MacCoU,  H. 
McCowan,  J.,  462 
Mean  Value,  theorem  of,  420 
Measurement,  41;  In  Projective  Geometry, 

294;   In   theory   of   irrationals,   397;   Of 

areas,  455 
Mechanics,  ig,  37,  171,  172,  179,  183,  211, 

229,  231,  240,  242,  255,  260,  276,  288,  296, 

307,  308,  338,  384,  391,  447-464,  477,  481; 

Theory  of  top,  458;  Fluid  motion,  ^60- 

464;  Least  action,  240,  255     See  Statics, 

Dynamics 
Mehler,  G.,  470 

Mehmke,  R.,  266,  366,  483,  484 
Mei  Ku-ch'6ng,  77 
Meissel,  E.,  416 
Meissner,  W.,  443 
Melanchthoii,  140 
Menaechmus,  27;  20,  39,  40,  107 
Mendizabel-Tamborrel,  M.  J.  de,  483 
Menelaus,  46,  47;  119;  Lemma,  46 
Mengoli,  P.,  173 
Mensuration  in  Euclid,  T,y,  in  Boethius,  67; 

in  China,  71 
M^ray,  C,  397;  400,  426 
Mercator,  G.,  189,  29s,  314 
Mcrcator,  N.,  156;  188,  206 
Meridian  measured,  200,  244;  Zero  merid- 
ian, 259 
Mersenne,  P.,  156,  163.  168,  174,  177,  181, 

183,  188;  M.  numbers,  167 
Mertens,  F.,  373,  43S,  439,  446 


504 


INDEX 


Meteorology,  462-464 
Method  of  exhaustion.   See  Exhaustion 
Method  of  fluxions,  194,  196.    See  Fluxions 
Method  of  tangents,  51,  163,  164,  177,  180, 

189,  193,  207,  209,  212;  Inverse  method 

of,  180,  207-209 
Metius,  A.,  73 

Metric  system,  256,  259,  265,  266 
Meusnier,  J.  B.  M.,  320 
Meyer,  A.,  384 
Meyer,  G.  F.,  372 
Meyer,  O.  E.,  461;  469,  478 
Meyer,  R.,  190 
Meyer,  W.  F.,  280,  300,  346 
Meziriac.    See  Bachet  de  Meziriac 
Michell,  J.,  230 

Michelson,  A.  A.,  465,  472,  479 
Mikami,  Y.,  71,  78-81,  88 
Mill,  J.  S.,  379;  173 
Miller,  G.  A.,  82,  279,  341,  350,  351-355, 

357-361 
Milner,  I.,  248 
Minchin,  G.  M.,  457 
Minding,  F.  A.,  414;  321 
Minimal  surfaces.     See  Surfaces,  minimal 
Minkowski,  H.,  480;  335,  371,  404,  441,  445, 

481 
Miran  Chelebi,  no 
Mirimanoff   D.,  442,  443 
Mises,  R.  M.  E.,  391 
Mitch tO.  H.  H.,  360 

Mittag-x^efBer,  G.  M.,  427;  279,  388,  426 
Mitzscherling,  A.,  437 
Miyai  Antai,  81 
Mobius,  A.  F.,  289;  287,  297,  310,  323,  336, 

337,  408,  423,  437,  449,  455 
Models  (geometric),  328,  329 
Modular  equations,  352,  416,  417 
Modulus,  first  use  of  term,  265 
Mohr,  0.,  296 
Moigno,  F.,  370;  241,  364 
Moivre,  A.  de.    See  De  Moivre 
Molenbroek,  P.,  335 
Molien,  Th.,  339 
Molk,  J.,  280 
Moller,  M.,  464 
Mollerup,  P.  J.,  327 
Mollweide,  K.  B.,  235;  437 
Moments  of  fluxions,  194 
Moments  of  quantities,  195,  196 
Momentum,  172 
Monge,  G.,  274,  275;  41,  231,  232,  246,  266, 

270,  276,  286,  287,  290,  296,  309,  315,  319, 

320,  322,  371,  384,  38s.  386 
Montmort,  P.  R.  de,  224;  222,  230,  383 
Montucla,  J.  F.,  250;  162,  185 
Moon,  theory  of,  105,  204,  240,  245,  259, 


260,  262,  449,  450,  451,  453,  462;  Libra- 

tion  of,  252;  Variation  of,  106 
Moore,  C.  L.  E.,  322 
Moore,  E.  H.,  318,  325,  357,  358,  394,  395, 

404,  405;  Quoted,  403 
Moore,  H.  L.,  380 
Moore,  R.  L.,  325,  328 
Morera,  G.,  428 
Mori  Kambei  Shigeyoshi,  78 
Morin,  486 

Moritz,  R.,  152,  345,  446,  447 
Morley,  E.  W.,  479 
Morley,  F.,  319,  320,  433 
Morley,  S.  G.,  69 
Mortality,  171 
Moschopulus,  M.,  128 
Moser,  L.,  381 

Motion,  laws  of,  171,  179,  199 
Moulton,  F.  R.,  327,  450,  453,  459,  46c; 

Quoted,  389,  454 
Mourraille,  J.  R.,  247,  269,  364 
Mouton,  G.,  206;  215 
Muir,  Th.,  340,  341,  484 
Miiller,  F.,  279 

Miiller,  J.    See  Regiomontanus 
Miiller,  J.  H.,  485 
Miiller,  R.,  301 
Multiple  points,  295 
Mu  Niko,  77 
Muramatsu,  79 
Musa  Sakir,  104 
Musical  proportion,  6 
Mydorge,  C,  166;  164 
Nachreiner,  V.,  341 
Nagelsbach,  H.,  341,  365 
Napier,  J.,   149-155;   127,   145,   146,   148; 

Analogies,  152;  Rule  of  circular  parts,  152 
Napier,  M.,  147 

Napoleon  I,  31,  268,  270,  275,  276 
Nascent  quantities,  192 
Nasir-Eddin,  108 
Nau,  M.  F.,  89 

Navier,  M.  H.,  465;  379,  383,  460,  461,  468 
Nebular  hypothesis,  260,  450 
Negative  numbers,  61,  71,  75,  93,  94,  107, 

123,  138,  141,  276,  289 
Neikirk,  L.  I.,  360 
Neil,  W.,  181;  188 
Nekrasoff,  P.,  365 
Nemorarius,  J.,  127;  118 
Neocleides,  28 

Nesselmann,  G.  H.  F.,  62,  in 
Netto,  E.,  341,  348,  354,  425 
Neuberg,  J.,  300;  his  circle,  300 
Neumann,  C,  470;  311,  321,  393,  430,  449, 

471.  472 
Neumann,  F.,  311, 313 


INDEX 


505 


Neumann,  F.  E.,  470;  468,  473.  474.  475.  477 

Neville,  E.  H.,  395 

Newcomb,  S.,  451;  308,  309,  383 

Newman,  F.  \V.,  484 

Newton,  I.,  191-205;  3,  42,  50,  80,  143,  161, 
182,  183,  189,  190,  219,  220,  222-224,  226, 
228,  229,  231,  232,  234,  239,  242-245, 
251,  257,  160,  262,  264,  273,  287,  303,  332, 
342,  361,  366,  369,  373,  458,  464,  481, 
Anagram,  213;  Approximation  to  roots, 
202,  227,  247,  269,  271,  364;  Binomial 
theorem,  186;  Controversy  with  Leibniz, 
212-218;  De  analyst  per  acqiialioncs,  192, 
196,  202,  212,  214;  Eminicratio  linearum 
tertii  ordinis,  204,  320;  Fractional  and 
negative  exponents,  178;  Gravitation 
Oaw  of),  199;  Least  resistance  (solid  of), 
201,  234,  284;  Method  of  Fluxions,  193- 
196,  202,  203;  Notation  of  fliLxions,  212, 
220;  Parallelogram  (Newton's),  203; 
Portsmouth  collection  of  MSS.,  200, 
204;  Principia,  196,  199,  200,  210,  213, 
220,  226,  234;  Problem  of  Newton,  201; 
Problem  of  Pappus,  176;  Quadrature  of 
Curves,  196,  197,  214;  Reflecting  telescope, 
204;  Rule  of  imaginary  roots,  202,  344; 
Scholium  {Prin.  II,  7),  213,  214,  216; 
Sextant,  204;  Similitude,  mechanical, 
457;  Universal  Arithmetic,  201,  235 

Newton,  J.,  152 

Nicolai,  F.,  437 

Nicole,  F.,  224;  227 

Nicolo  of  Brescia.    See  Tartaglia 

Nicomachus,  58;  48,  59,  67 

Nicomedes,  42 

Nieuwentijt,  B.,  218 

Nievenglowski,  B.  A.,  241 

Nine-point-circle,  298 

Nippur,  librarj'  at,  4,  7 

Niven,  C,  473 

Niven,  W.  D.,  470 

Noble,  C.  A.,  430 

Noether,  M.,  136-138,  295,  296,  313,  314, 
316,  319,  340,  354.  419,  430,  431 

Nomography,  481,  482 

Non-euclidean  geometry,  32,  302-309;  and 
relativity,  481 

Nonius.    See  Nunez 

Norlund,  N.  E.,  391 

Norwood,  R.,  158 

Notation  of  Algebra  and  Analysis:  Abridged 
in  analytics,  310;  Arabs,  early,  100; 
Arabs,  late,  iii;  Algebraic  forms,  345, 
346;  Algebraic  equations,  247;  Calculus, 
206,  207,  249,  272;  Chinese,  76;  Con- 
tinued fractions,  239;  Determinants, 
340,  341;  "Function  of,"  234;  Greek,  125; 


Infinity,  1S5;  Hindu,  93,  125;  Logic,  410; 
Multiplication,  157,  158;  Ratio,  157; 
Renaissance,  125-127,  139,  156,  157; 
Trigonometry,  158,  234,  272;  Vector 
analysis,  334;  Symbols  used  by  Diophan- 
tus,  61,  by  Oughtred,  157;  by  Leibniz, 
211;  Symbols  +  and  — ,  139,  140;  >  and 
<,  157;  nl,  341;  identity  =,  342;  (),  158; 
summation  S,  235;  i  for  V-i.  235;  e  = 
2.718.  .  .,  234;  =,  140;  -j-,  140;  V.  140; 
IT  =3.14159.  158;  X,  157;  ::.i57;  v^, 
157;  a^  178;  a?,  178;  a°,  192 

Notation  of  arithmetic:  Fractions,  12,  65; 
Decimal  fractions,   148;  Proportion,  211 

Notation  of  Geometry:  Similarity,  211; 
Congruence,  211 

Notation  of  Numbers:  Babylonian,  4-6; 
Egyptian,  11;  Chinese,  72,  77;  Hindu,  88- 
90;  Maya,  6g,  70;  Greek,  52,  53;  Roman, 
63 

Notation,  principles  of,  4,  11 

Nother,  M.    See  Noether,  M. 

Nozawa,  79 

Number-fields,  444-446 

Number-mysticism,  7,  55,  56,  144,  145.  See 
Magic  squares 

Number-systems,  70.  See  Notation  of 
numbers 

Numbers:  Algebraic,  446,  447;  Amicable,  56, 
104,  109,  239;  Complex,  442;  Cardinal, 
403,  447;  Cubic  residues,  414,  437;  Bi- 
quadratic residues,  437;  Quadratic  res- 
idues, 435;  Concept  of  22;  Defective,  56; 
Excessive,  56;  Heteromecic,  56;  Ideal, 
442;  Negative,  61,  71,  75,  93,  94,  107,  123, 
138,  141,  276,  289;  Ordinal,  403,  447; 
Partition  of  239,  344,  367,  444;  Perfect, 
56,  104,  114,  167;  Prime,  58,  167,  169, 
239,  249,  254,  344.  438,  439;  Pentagonal, 
16S;  Polygonal,  168;  Irrational,  2,  19, 
22,  32,  43,  57,  61,  86,  93,  94,  103,  133,  140, 
330,  396-400,  483;  Transcendental,  143, 
440,  446,  447;  Transfinite,  426;  Trian- 
gular, 56,  168,  173 

Numbers,  theory  of,  48,  59,  106,  114,  124, 
2S5,  344,  348,  362,  434-446;  Euler,  168- 
170,  239;  Fermat,  167-169;  Fields,  444, 
445;  Lagrange,  254;  Legendre,  266,  267; 
Law  of  quadratic  reci  procity,  435;  Law 
of  large  numbers,  222,  380.  See  Last 
theorem  of  Fermat,  Magic  squares. 

Numerals.  See  Hindu-Arabic  numerals. 
Notation  of  numbers 

Numerical  equations,  solution  of,  74,  75" 
77,  no,  202,  203,  227,  247;  Chinese,  74, 
76,  77,  iii;  Continued  fractions,  254; 
Dandelin,    364,    365;    Gralle.    364-366; 


^o6 


INDEX 


Homer,  74,  75,  271,  365;  Infinite  fleter- 
minants,  ,365;  Infinite  series,  227,  365; 
Japanese,  80,  81;  Leonardo  of  Pisa,  124; 
Newton,  203;  Raphson,  203;  Recent  re- 
searches, 363-366.  See  Algebra,  Equa- 
tions, Roots 

Nunez,  .P.,  142;  143 

Obenrauch,  F.  J.,  297 

Oberbeck,  A.,  464 

Ocagne.    See  D'Ocagne 

Odhner,  W.  T.,  485 

Oenopides,  17;  15 

Ohm,  G.  S.,  281 

Ohm,  M.,  329;  330,  424 

Ohrtmann,  C,  278 

Olbers,  H.  W.  M.,  452;  435,  437,  447, 
448 

Oldenburg,  H.,  178,  187,  212-214,  215 

Olivier,  T.,  296 

Omar  Khayyan;,  103,  107 

Oppel,  F.  W.,  235 

Oppert,  8 

OppikoSer,  J.,  486 

Oppolzer,  T.  v.,  452;  455 

Orchard,  155 

Ordinate,  175 

Oresme,  N.,  127;  148 

Origen,  67,  126 

Orontius,  116 

Orthocenter,  297 

Oscillation,  center  of,  183 

Osculating  curves,  211 

Osgood,  W.  F.,  372,  40s,  433 

Ostrogradski,  M.,  369,  371,  456 

Otho,  v.,  73,  132 

Otto,  V.    See  Otho,  V 

Oughtred,  W.,  157-15Q.  ^n,  i37,  148,  152, 
153,  155,  174,  192,  481 

Ovals  of  Descartes,  176 

Ovidio,  E.  d',  308 

Ozanam,  J.,  170,  155 

T,  approximations  to,  7,  10,  35,  71,  73,  77, 
79-81,  87,  186,  206,  238,  104,  143,  483, 
486;  Determination  of,  17,  185,  225,  238; 
Notation  of,  158;  Proved  irrational,  246, 
268;  Proved  transcendental,  2,  143,  362, 
440,  446 

Pacioli,  L.,  128-130;  125,  133,  141,  144,  146 

Pad6,  H.,  375 

Padmanabha,  85 

Padoa,  A.,  quoted,  410 

Pagani,  G.  M.,  330 

Painleve,  P.,  279,  389,  453,  454 

Pajot,  L.  L.,  170 

Palatine  anthology,  59,  60 

Paolis,  R.  de,  308 

Fapperitz,  E.,  286 


Pappus,  49,  50;  21,  30,  33,  41,  42,  4S,  54.  55. 
142,  166;  Problem  of  50 

Parabola,  162,  177,  185,  188,  206,  224: 
Cubical,  1 82,  188,  208;  Divergent,  204, 
244;  Semi-cubical,  181;  Focus  of,  50 

Paradoxes,  400,  409.    See  Zeno 

Parallel  lines,  166,  302,  303,  327;  Defined, 
48.  See  Parallel  postulate,  Euclid,  Non- 
euclidean  geometry 

Parallel  motion,  300 

Parallel  postulate,  32,  48,  108,  184,  302,  303, 
305,  308;  "proofs"  of,  48,  108 

Parallelogram  of  forces,  172 

Parent,  A.,  167 

Paris  academy  of  sciences,  168,  182,  246 

Parmenides,  24 

Parseval,  M.  A.,  376 

Partial  differential  equations,  196,  242,  251, 
255,  263,  264,  270,  275,  281,  392,  313, 
355.  384-388,  422,  456 

Partition  of  numbers,  239,  344,  367,  444 

Pascal,  B.,  164-166;  76,  142,  146,  147,  163, 
162,  167,  180,  183,  184,  187,  190,  206,  207, 
246,  272,  273,  287,  311,  485;  Pascal  line, 
290;  On  chance,  170;  Theorem  on  hexa- 
gon, 228,  166,  318,  327;  Calculating  ma- 
chine, 165,  485 

Pascal,  Ernesto,  318,  319,  340,  371,  486; 
Quoted,  371 

Pasch,  M.,  309,  399,  400,  409 

Pavanini,  G.,  453 

Pascal,  Etienne,  164 

Peacock,  G.,  273;  122,  125,  272,  145,  330, 
332;  Principle  of  permanence,  273,  337 

Peano,  G.,  285,  289,  309,  325,  326,  348,  387, 
400,  401,  407-409;  Formulaire,  408 

Pearson,  K.,  380,  381,  383,  465,  468 

Peaucellier,  A.,  301 

Pedal  curves,  228 

Peirce,  B.,  338;  278,  285,  286,  332,  383,  4Si. 
457;  Linear  associative  algebra,  338,  339 

Peirce,  C.  S.,  407;  31,  285,  309,  337,  338,  33Q 

Peletier,  J.,  137,  156 

PeU,  A.  J.,  396 

Pell,  J.,  169;  206 

Pell's  equation,  96,  169 

Pemberton,  H.,  219;  192,  199,  220 

Pendulum,  183,  205,  266,  460,  478 

Pendulum  clocks,  183 

Pentagram,  19 

Pepin,  Th.,  436 

Perfect  numbers,  56,  104,  114,  167 

Periodicals  (mathematical).    5ee  Journals 

Permutations,  221.    See  Probability 

Pernter,  J.  M.,  463 

Perrault,  C,  182 

Perron,  O.,  348,  370 


INDEX 


507 


Perseus,  42 

Persons,  W.  M.,  380 

Perspective,  166,  227.  See  Projective 
geometry,  Descriptive  geometry 

Perturbations  of  planets,  240  252,  261; 
Scf  Astronomy 

Pesloiian,  C.  L  de,  4x2 

Peters  A.,  324 

Peters,  J.,  483 

Petersburg  problem,  223,  243 

Peterson,  J.,  436;  323 

Petrus  Ilartqngius,  81 

Peurhach,  G.,  127;  131,  132 

Pezzo,  Del,  307 

PfaEf,J.F.,  384:434,  435 

Pliilalethes  Cantabrigiensis,  219.  See 
Jurin,  J. 

Philippus  of  Mende,  29 

Philolaus,  19;  25,  55 

Philonides,  39 

Phragmen,  E.,  426,  427,  453 

Piazzi,  G ,  447 

Picard,  C.  E.,  433;  279,  281,  315-317,  35i, 
387,  38S,  391,  413,  415.  416.  432;  Quoted, 
258,  264,  353,  429,  447,  454,  477;  Inte- 
grals, 317 

Picard,  J.,  200 

Picone,  M.,  391 

Picquet,  H.,  341 

Piddington,  H.,  463 

Pieri,  M.,  327;  309,  328,  400,  409 

Pierpont,  J.,  406,  351;  Quoted,  425,  430,  431 

Pincherle,  S.,  394,  405 

Piola,  G.,  467 

Pitcher,  A.  D.,  395 

Pitigianis,  F.  de,  126 

Pitiscus,  B.,  132;  137,  148 

Pizzetti,  P.,  382 

Plana,  G.  A.  A.,  449;  465,  473 

Planetesimal  hypothesis,  450 

Planimeters,  486 

Planisphere,  48 

Planudes,  M.,  128 

Plateau,  J.,  371;  461 

Plato,  25-29;  7,  15,  19,  21,  23,  30,  59;  In- 
scription at  his  academy,  2,  26;  Quoted, 
9,  IS 

Plato  Tiburtinus  (Plato  of  Tivoli),  105,  118, 
123 

Platonic  figures,  35 

Platonic  number,  7 

Platonic  school,  25-29 

Playfair,  J.,  145,  192,  218,  302 

Pulcker,  J.,  309-312;  278,  288,  297,  3o6, 
313,  314,  335.  354;  P-  equations,  310;  P. 
lines,  290 

Plutarch,  15,  16,  34 


Pohlke,  K.,  296 

Poignard,  170 

Poincare,  II.,  388-391;  327,  339,  34i,  353, 
355,  361,  375,  378,  386,  387,  393,  401,  402, 
41S,   429.   432,   433,  438,  450,   452,   453, 

454,  462,  477,  479 
Poinsot,  L.,  455 ;  293,  379,  458 
Point,  26 

Point  sets,  325,  326,  394,  395,  404;  Denu- 
mcrable,  403;  Non-measurable,  403.  See 
Aggregates 

Poisson,  S.  D  ,  465-467;  164,  223,  293,  340, 
369,   377.   379,   380,   383,   413,   438,   440, 

455,  456,  458,  460,  461,  464,  468,  470, 
472,  473 

Polar  coordinates,  221,  224 

Polars,  theory  of,  167 

Pole,  W.,  383 

Polenus,  4S5 

Polya,  G.,  362 

Polyhedra,  240 

Poncelet,  J.  V.,  287,  288;  166,  190,  268,  275, 
276,  286,  290,  297,  298,  301,  308,  311, 
354,  395,  467;  P-  paradox,  310 

Poor,  V.  C,  335 

Porism,  33 

Porphyrius,  7,  45 

Posidonius,  48 

Position,  principle  of.    See  Local  value 

Postulates,  35;  complete  independence  of, 
395;  Of  Euclid,  31;  Of  geometry,  326- 
328.    See  Parallel  postulate 

Potential,  256,  263,  264;  2S4,  314,  389,  sgs, 
394,  419,  422,  472,  477 

Potron,  M.,  360 

Pouchet,  481 

Powell,  B.,  260 

Power  series,  185,  387,  420,  431,  445  Sei 
Series 

Poynting,  J.  H.,  474,  475 

Prandl,  L.,  334 

Precession  of  equinoxes,  242 

Prestet,  J.,  248;  170 

Preston,  T.,  476 

Prihonski,  F.,  367 

Prime  numbers,  58,  167,  230,  249,  254,  344, 
438,  439;  Fermat,  160;  Prime  number 
theorem,  439 

Prime  and  ultimate  ratios,  189,  257 

Principia  of  Newton.    See  Newton 

Principle  of  duality.    See  Duality 

Principle  of  position.    See  Local  value 

Pringsheim,  A.,  374,  400,  475,  428 

Probability,  165,  170,  171,  183,  221-224, 
229,  230,  240,  243,  244,  258,  2f)2,  263,  273, 
344,  367,  377-383,  478;  Inverse  pro- 
bability, 2jO,  37S;  Local  probability,  230. 


5o8 


INDEX 


378,  37v;  Rforal  expectation,   223,  378; 

Problem  of  points,  170,  224,  263;  Law  of 

large  numbers,  222,  380 
Problems  for  quickening  tiie  mind,  114 
Problem  of  Pappus,  176 
Problem  of  three  bodies.    See  'I'hree  bodies 
Proclus,  si;  15,  17,  21,  28,  30,  31,  33,  42, 

44,  48,  49,  142,  302 
Probleme  des  rencontres,  366 
Progression.    See  Arithmetical;  Geometrical 
Projection:   Stereographic,  48,   167;  Ortho- 
graphic, 48;  Globular,  167 
Projective   geometry,    276,    285,    292-294, 

297,  308,  327,  328 
Prony,  F.  M.  de,  300,  301 
Prony,  G.  Riche  de,  482 
Proportion,  6,  10,  16,  19,  20,  22,  31,  32,  56, 

58,   73,   75;   Euclid's  theory,  32;  Arith- 
metical, 56;  Geometrical,  56;  Harmonic, 

56;  Musical,  56 
Prym,  F.,  418 
Pseudo-sphere,  305 
Ptolemy,  46-48;  j,  7,  45,  87,  g6,  loi,  102, 

105,  100,  127,  129-131,  160,  184,  306,  314; 

Almagesi,  5,  46,  49,  50,  54,  88,  100,  10 1, 

104,    119,    120;    PtoleLnaic    system,    46; 

Tables  of  chords,  47 
Puchta,  A.,  340  ' 

Puiseux,  V.  A.,  420 
Pulverizer,  95 
Purbach,  G.,  127;  131,  132 
Pyramids  of  Egypt,  9,  10,  14,  16 
Pythagoras,  17-20;  2,    15,   55,   57,  68,  80, 

104 
Pythagorean  school,  17-19 
Pythagorean  theorem,  2,  18,  30,  86-88,  97; 

Nicknames  of,  129 
Pythagoreans,  7,  31,  239 
Quadrant,  centesimal  division  of,  259.    See 

Degree 
Quadratic  equations,    13,   57,    72,   74,   94; 

Hindu  method  of  solving,  94 
Quadratic  reciprocity,  239,  267 
Quadratrix,  21,  28,  49 
Quadratura  curvarum  (Newton's)  196,  197, 

198 
Quadrature  of  curves,   181,   184,   192,  206, 

207 
Quadrature  of  the  circle,  i,  2,  17,  74,  79,  133, 

143,  169,  181,  182,  185,  212,  236,  246,  446; 

Impossibility,  i,  2,  143,  362,  440,  446 
Quadrivium,  113 
Quantity,  285,  396,  398 
Quaternions,  307,  323,  33Q    332-335,  337, 

353,  473;  Quaternion-Ass' n,  335 
Quercu,  a,  143 
Querret,  J.  J,  273 


Quetelet,  A.,  380;   144,  148,  378;  A\erage 

man,  380 
Quetelet,  L.  A.  J.,  289 
Raabe,  J.  L.,  374;  397 
Radau,  R.,  452 

Radians,  483 ;  Origin  of  word,  484 
Radius  of  curvature,  196,  221 
Radon,  J.,  406 
Rany,  L.,  Si$ 
Rahn,  J.  H.,  140;  lu^ 
Rallier  des  Ourmes,  170 
Ramanujan,  S.,  367 
Ramus,  P.,  142 
Rangacarya,  85 
Rankine,  W.  J.  M.,  476;  468 
Ranum,  A.,  322 

Raphson,  J.,  203;  227,  247,  271,  364 
Rational,  origin  of  word,  68 
Rawlinson,  R.,  234 

Rayleigh,  Lord,  464;  448,  461,  462,  465,  4711. 
Reciprocal  polars,  288,  296 
Reciprocal  radii,  392 
Recorde,  R.,  140;  146 
Recurring  series,  227,  230 
Redfield,  W.  C.,  462 
Reductio  ad  absurdum,  25 
Reech,  F.,  457 
Regiomontanus,    131;    iii,    132,    139,    141, 

143,  14s,  146,  147 
Regnault,  H.  V.,  473,  476 
Regula  falsa,  12,  13,  91,  93,  103,  137,  366; 

Double,  44,  103,  no,  123 
Regula  sex  quantitatum,  46,  47,  109 
Regular  solids,  18,  27,  29,  a,  43,  106,  159, 

347 
Reid,  A.,  155 
Reid,  W.,  463 
Reiff,  R.,  238 
Reiss,  M.,  341 

Relativity,  principle  of,  335,  479-481 
Resal,  H.,  455 
Residual  analysis,  247 
Resolvents,  253 
Resultant,  249 
Revue  semestrielle,  278 
Reye,  T.,  294,  464 
Reymer,  178 
Reynolds,  O.,  462 
Rhaetius,  131,  132;  483 
Rhind  collection,  9 
Riccati,    J.    F.,    224;    223,    247;    Riccati's 

differential  equation,  223,  225 
Riccati,  V.,  247 
Ricci,  G.,  333,  356 
Ricci,  M.,  77 

Richard,  J.  A.,  401,  402,  409 
Richard  of  Wallingford,  128 


INDEX 


509 


Richardson,  L.  J.,  68 

Richelot,  F.  J.,  417;  311,  313.  43^ 

Richmond,  H.  W.,  318,  319 

Riemann,  G.  F.  B.,  421-423;  258,  279,  284, 
306,  307,  313,  314.  316,  324,  342,  375,  376 
38s,  386,  387,  398,  400.  405.  418,  419,  428, 
429,  430,  431,  432,  433,  438,  439,  445.  462; 
Riemann's  surface,  307,  347,  417,  422, 
426,  470,  474;  Zeta-function,  439 

Riesz,  F.,  37O,  396,  406 

Riesz,  M.,  427 

Rietz,  H.  L.,  357,  360 

Right  triangle,  169.    5ftf  Triangle 

Rithmomachia,  116 

Ritter,  E.,  433 

Robb,  A.  A.,  480 

Robert  of  Chester,  119 

Roberts  S.,  301,  302 

Roberts  W.,  314 

Roberval,  G.  P.,  162;  42,  146,  163,  164,  165, 
176,  177,  17S,  180,  iSi,  183,  190, 191 

Robins,  B.,  219;  220,  369,  458 

Roch,  G.,  431 

Rodriques,  O.,  469 

Roe,  N.,  152 

Rogers,  R.  A.  P.,  313 

Rohn,  K.,  319 

RoUe,  M.,  224;  220;  RoUe's  theorem,  224; 
Method  of  cascades,  224 

Roman  notation  and  numerals,  63,  178 

Romanus,  A.,  143;  133,  138,  144 

Romer,  O.,  190 

Roots,  80,  106,  III,  115,  123,  141,  440; 
Chauchy's  theorem,  363;  Cube  root,  71, 
74,  123;  Equal  roots,  180;  Every  equa- 
tion has  a  root,  349;  Imaginary,  123,  135, 
156,  179,  202,  248,  249,  363-365;  In 
Galois  theory,  351;  Irrational,  43,  61,  103; 
Negative  roots,  61,  107,  135,  141,  156; 
Square  root,  54,  71,  94,  iii;  Sturm's 
theorem,  363;  Upper  and  lower  limits, 
180,  202,  225,  269,  361.  See  Equation, 
Irrationals,  Negative  numbers 

Rosenhain,  J.  G.,  418;  414,  417 

Ross,  R.,  383 

Rossi,  C,  365 

Rothe,  R.,  322 

Rothenberg,  S.,  384 

Rouchg,  E.,  393 

Rougier,  L.,  481 

Roulettes,  81,  167,  224 

Routh,  E.  J.,  457,  458,  474 

Rover,  W.  H.,  459;  Quoted,  460 

Rowland,  H.  A.,  472;  461,  474,  475 

Royal  Society  of  London,  founded,  184 

Royce,  J.,  410 
Radio,  F.,  17,  51 


RudolfT,  Ch.,  140 

Ruffini,  P.,   349;    75,    271,   350,   352,   353,' 

RufSni's  theorem,  350 
Riihlmann,  R.,  477 

Rule  of  false  position.    See  False  position 
Rule  of  three,  93,  103 
Ruled  surfaces,  295 

Ruler  and  compasses.    See  Constructions 
Runge,  C,  426,  427 
Runkle,  J.  U.,  451 
Russell,  B.,  285,  328,  399,  400,  402,  407, 

409 
Saccheri,  G.,  304;  184,  302,  305 
Sachse,  A.,  375 
Sacro  Bosco,  129 
Safford,  T.  H.,  451 
Saint-Venant,  B.  dei,  468;  337,  350,  461,  466. 

467,471 
Saladini,  G.,  221 
Salmon,  G.  (1819-1904),  290,  297,  312,  313, 

317,    320.    342,    345,    346,    361;    Salmon 

point,  291 
Salvis,  A.  de,  172 
Sand-counter,  54,  78,  90 
Sang,  E.,  482,  486 
Sangi  pieces,  76,  78,  80 
Sarasa,  A.  A.  de,  181,  188 
Sarrau,  E.,  471 
Sarrus,  P.  F.,  369;  301,  370 
Sarton,  G.,  389 
Sato  Seiko,  79 
Saturn's  rings,  451 
Saurin,  J.,  224;  143 
Sauveur,  J.,  170,  367 
Savart,  F.,  465 
Savasorda,  A.,  121,  123 
Sawaguchi,  79 
Sawano  Chuan,  81 
Scaliger,  J.,  143 
Scarpio,  U.,  357 
SchefTers,  G.  W.,  339,  355 
SchefHer,  H.,  367 
Schellbach,  C.  H.,  291 
Schepp,  A.,  458;  371,  377 
Schering,  E.,  436;  308,  421 
Scherk,  H.  F.,  340 
Scheutz,  485 
Schiaparelli,  G.  V.,  382 
Schilling,  M.,  328 
Schimmack,  R.,  405 
Schlafli,  L.,  417;  308,  318,  376,  47° 
Schlegel,  V.,  337;  309,  336 
Schlesinger,  J.,  296 
Schlesinger,  L.,  387 
Schlick,  O.,  458 
Schlomilch,  O.,  449;  365 
Schmidt,  C,  383 


510 


INDEX 


Schmidt,  E.,  341,  393,  394 

Schmidt,  F.,  303 

Schmidt,  W.,  44 

Schone,  H.,  44 

Schonemarin,  P.,  348 

Schonflies,  A.,  326,  400,  401 

Schottenfels,  I.  M.,  359 

Schottky,  F.,  425,  431 

Sclireiber,  G.  296;  276 

Schreiber,  H.    See  Grammateus 

Schroder,  E.,  407;  285,  365,  405,  408 

Schroter,  H.,  466;  314,  417 

Schubert,  F.  T.  v.,  348 

Schubert,  H.,  293 

Schulze,  J.  K.,  247 

Schumacher,  H.  C,  411,  437,  449 

Schur,  F.,  327 

Schuster,  A.,  470 

Schutte,  F.,  176,  177,  183,  245,  275 

Schwarz,  H.  A.,  431-432;  293,  319,  347, 
359,  368,  370,  372,  376,  390,  391,  428; 
Schwarzian  derivative,  432 

Schweikart,  F.  K.,  305 

Schweins,  F.,  340 

Schweitzer,  A.  R.,  395 

Scott,  R.  F.,  341 

Scotus,  Duns,  126 

Sebokht,  S.,  89 

Section,  golden,  28 

Seeber,  L.,  444 

Seelhoff,  P.,  167,  445 

Segner,  J.  A.,  248;  458 

Segre,  C,  289,  307,  318,  322,  431 

Seguier,  J.  A.  de,  360 

Seidel,  P.  L.  v.,  377 

Seitz,  E.  B.,  379 

Seki  Kowa,  80;  81 

Selling,  E.,  444 

Sellmeyer,  W.,  471 

Serenus,  45 

Series,  75,  77,  80,  81, 106, 127,  172, 181,  187, 
188, 192,  196,  206,  212,  227,  232,  238,  246, 
248,  257,  258,  361,  367,  373,  425,  434,  411; 
Alternating,  373;  Asymptotic,  375;  Con- 
ditionally convergent,  374;  Convergence 
of,  227,  249,  270,  284,  367,  373-37S,  417; 
Divergent,  375,  454;  Hypergeometric, 
185,  387.  432;  Product  of  two  series,  373, 
374;  Of  reciprocal  powers,  238;  Power- 
series,  185,  387,  420,  431,  445;  Trigo- 
nometric series,  419,  431;  Uniform  con- 
vergence, 84,  377;  Recurrent,  127.  See 
Arithmetical  progression,  Geometrical 
progression 

Serret,  J.  A.,  314;  319. 352  384,  385,  436,  456 

Serson,  458 

Sc-vant,  M.  G.,  325,  375 


Servois,  P.,  273,  275,  2S8 

Sets  of  curves,  405 

Sets  of  lines,  405 

Sets  of  planes,  405 

Sets  of  points,  325,  326,  394,  395,  404 

Seven,  F.,  293,  317,  319 

Sexagesimal  numbers,  4,  5;  43,  47,  88,  100, 

483;  Fractions,  5,  54,  483;  Invention  of, 

5,6 
Sextant,  204 
Sextus  Empiricus,  48 
Sextus  Juhus  Africanus,  48 
Shades  and  shadows,  297.    See  Descriptive 

geometry 
Shakespeare,  igo 
Shanks,  VV.,  206 
Sharp,  A.,  206;  24 
Sharpe,  F.  R.,  319 
Shaw,  H.  S.  Hele,  486 
Shaw,  J.  B.,  sii,  339,  410 
Sheldon,  E.  W.,  372 
Shotoku  Taishi,  78 
Siemens,  W.,  463 
Silberstein,  L.,  479 
Simart,  G.,  316 

Similar  polygons,  19,  22,  32,  184 
Similitude,  mechanical,  457 
Simony,  O.,  323 
Simphcius,  51;  22,  23,  48,  184 
Simpson,  T.,  235;  227,  234,  244,  365,  382 
Simson,  R.,  277;  31,  2ii 
Sindhind,  gg 
Sine  function,   104,  no;  Origin  of  name, 

105,  iig 
Singhalesian  signs,  89 
Singular  solutions,  211,  224,  227,  239,  245, 

254,  255,  264,  383 
SinigaUia,  L.,  395 
Sisam,  C.  H.,  322 
Slide  rule,  158,  159 
Slobin,  H.  L.,  446 

Sluse,  R.  F.  de,  180;  42,  188,  208,  209 
Sluze.    See  Sluse 
Smith,  A.,  457 
Smith,  D.  E.,  7,  68,  71,  78,  86,  88,  89,  116, 

121,  128,  177,  184,  291,  332 
Smith,  H.  J.  S.,  441;  342,  416,  438,  442,  444 
Smith,  R.,  226 
Smith,  St.,  292 
Snellius,  W.,  143 
Sniadecki,  J.  B.,  258 
Snyder,  V.,  320 

Societies  (mathematical),  279,  296 
Socrates,  25 
Sohncke,  L.  A.,  417 
Solar  system  (stability  of),  ^84 
Solids,  regular,  19,  a,  106,  159 


INDEX 


511 


5)o!iclu5,  foi  writing  fractions,  332 

Summer,  J.,  445 

Sommerfeld,  A.,  458 

Sommerville,  D.  M.  Y.,  305,  306,  329 

Somoff,  J.  I.,  457 

Sophists,  20-25 

Soroban,  78 

Sosigenes,  66 

Sound,  240,  251,  460,  464-470;  Velocity  of, 
264 

Space  of  n  dimensions.  See  geometry,  n 
dimensions 

Sparre,  Comte  de,  459 

Specific  gravity,  37 

Speidell,  J.,  152;  158 

Sperry,  E.  A.,  458 

Sphere,  19,  27,  33,  36,  42.  45.  5°,  79.  106, 
107,  314 

Spherical  harmonics,  232,  263,  469 

Spherical  trigonometry.    See  Trigonometry 

Sphero-circle  (imaginary),  293 

Spheroid,  36;  Attraction  of,  200,  267 

Spirals  of  Archimedes,  36,  50;  Fermatian, 
224;  Spherical,  50 

Spitzer,  S.,  369;  365 

Spottiswoode,  341;  337;  Quoted,  281 

Square  root.    See  Root 

Squaring  the  circle.  See  Quadrature  of  the 
circle 

S'ridhara,  85,  94 

StabiUty  of  solar  system,  260,  262 

Stiickel,  P.,  184,  238,  426 

"Stade,"  24 

Stager,  H.  VV.,  353 

Stahl,  H.,  30S 

Star- polygons,  127 

Statics,  171,  172,  181,  255,  282,  289;  Theory 
of  couples,  455.  See  Graphic  statics. 
Mechanics 

Statistics,  377-383;  Arithmetic  mean,  381, 
382;  Averages,  theory  of,  381;  Normal 
curve,  382;  Median,  381;  Frequency 
curve,  383;  Mode,  381;  Mortality,  381; 
Population,  381;  In  biology,  381;  Stand- 
ard deviation,  382 

Staudt,  von,  294-308;  280,  287,  297,  309, 
310 

St.  Augustine,  67 

Steele,  W.  J.,  457.  459 

Stefano,  A.  B.,  461 

Steiger,  O.,  485 

Steiner,  J.,  290-292;  287,  289,  297,  309, 
310,  311,  312,  313,  317,  318,  320,  323,  336, 
346,  362,  370,  411.  421,  423.  424;  Steiner 
point,  290;  Steiner  surface,  319 

Stekloff,  VV.  A.,  393 

Stephanos,  C,  289;  348 


Stereographic  projection,  48 

Stern,  M.  A.,  364;  365,  421,  436 

Sterneck,  R.  v.,  439 

Stevin,  S.,  147;  127.  i37.  148,  171.  178,  187 

Stewart,  M.,  277 

Stieltjes,  T.  J.,  375;  406,  446,  470 

Stifel,  M.,  140;  139,  141,  144-146,  149,  183, 
187 

Stirling,  J.,  229;  204,  229,  377 

St.  Laurent,  T.  de,  273 

Stokes,  G.  G.,  460,  461;  281,  284,  332,  377, 
461,  462,  465,  466,  468,  471,  47S 

Stolz,  O.,  42s;  35,  368,  399,  400 

Stone,  E.  J.,  382 

Story,  W.,  308,  323.  348 

Stoufifer,  E.  B.,  322 

Strassmaier,  P.  J.  R.,  8 

Stratton,  S.  W.,  465 

Strauch,  G.  W.,  370 

Stringham,  W.  I.,  308;  309 

Stroh,  E.,  348 

Stromgren,  F.  E.,  453 

Strutt,  J.  W.    See  Rayleigh,  Lord 

Struve,  G.  W.,  437 

Stubbs,  J.  W.,  292 

Stiibner,  F.  W.,  248 

Study,  E.,  289,  293,  308,  335.  339.  348.  356 

Sturm,  C,  363;  166,  269,  273,  310,  344,  420, 
456,  457,  472,  473;  Sturm's  theorem,  363, 
364.  366 

Sturm,  R.,  317;  295.  3i8,  319 

St.  Venant,  B.  de,  297 

St.  Vincent,  Gregory,  181;  182,  188,  190, 
206 

Suan-pan,  52,  76,  78 

Substitutions,  theory  of,  281,  417;  Orthog- 
onal, 342 

Sulvasiitras,  84-86 

Sumerians,  4 

Sundman,  K.  F.,  452,  454 

Sun-Tsu,  72;  73,  75,  78 

Surfaces,  49.  235,  275.  295.  296,  314-318, 
321,  322;  Anchor-ring,  323;  Confocal, 
293;  Cubic,  29s,  313.  314.  317.  318,  329. 
343;  Deformation,  321;  Isothermic,  314; 
Kummer,  418,  328;  Minimal,  315,  325. 
355,  371,  38s.  432;  Of  negative  curvature, 
307;  Plectoidai.  50;  Polar,  307;  Pseudo- 
spherical,  321;  Quintic,  319;  Quartic,  329; 
Ruled,  29s,  319,  320;  Third  order,  291, 
318;  Fourth  order,  314,  318,  319;  Second 
degree,  275,  290;  Second  order,  235,  295, 
311;  Universal,  296;  Wave-surface,  311, 
314,  319,  33^ 

Surveying,  44,  66,  77 

Surya  siddhanta,  84 

Siissmilch,  J.  P.,  380 


512 


INDEX 


Suter,  H.,  104,  log,  181 

Swan-pan.    See  Suan-pan 

Swedenborg,  E.,  260 

Sylow,  L.,  354;  414,  352,  357,  361;  Sylow's 

theorem,  354 
Sylvester,  J.  ].,  343-34Q;  202,  249,  27S,  282, 

28s,  2Q7,  312,  313,  317,  323,  324,  S3S,  334, 

340,  341,  342,  351,  361,  363,  379,  432,  418, 

438,  441,  444,  455,  472;  Link-motion,  301; 

Partitions,  344;  Reciprocants,  344 
Symbolic  logic,  205,  246 
Symmedian  point,  299 
Symmetric  functions,  235,  414 
Synthesis,  27 
Synthetic    geometry,    166,    167,    286-309. 

See  geometry.  Projective  geometry 
Syrianiis,  51 
Syzygies,  348 

Tabit  ibn  Korra,  104;  loi,  no 
Tables,  mathematical,  482-484 
Taber,  H.,  339,  340 
Tait,  P.  G.,  459;  272,  279,  322,  323,  333, 

334.  335.  337,  382,  457,  466,  470,  473,  476; 

Golf-ball,  459 
Takebe,  80;  81 
Talbot,  H.  F.,  413 
Tanaka  Kisshin,  79;  81 
Tangents,  method  of.     See  Method  of  tan- 
gents 
Tannery,  J.,  385;  314,  401,  351,  387,  433 
Tannery,  P.,  401;  7,  24,  32,  43,  45,  46,  53, 

60,  96,  177,  400;  Quoted,  175 
Tartaglia,  133,  134;  139,  141,  142,  146,  158, 

170,  183 
Taurinus,  F.  A.,  305;  184 
Tautochronous  cun,'e,  183,  413 
Taylor,  B.,  226;  155,  218,  229.  242,  245,  251; 

His  theorem,  226,  251,  257,  258,  394,  369, 

38s.  422 
Taylor,  H.  M.,  300;  His  circle,  300 
Tedenat,  273 
Teixeira,  G.,  320 
Telescope,  183;  Reflecting,  204 
Tenzan  method,  80 
Terquem,  O.,  228 
Teubner,  B.  G.  (firm  of),  328 
Thales,  15-18 
Theaetetus,  28;  30,  31,  57 
Theodorus  of  Gyrene.  57 
Theodosius,  44,  45,  104,  1 18-120 
Theon  of  Alexandria,  50;  31,  42,  43,  45,  54, 

67 
Theon  of  Smyrna,  59;  45,  48,  142 
Theorem  of  Pythagoras.    See  Pythagorean 

theorem 
Theory  of  numbers.  See  Numbers,  theory  of 
Theudius,  28,  30 


Thiele,  T.  N.,  378 

Thomae,  H.,  400 

Thomae,  J,  416 

Thoman,  155 

Thomas,  A.,  127,  182 

Thomas,  Ch.  X.,  485 

Thomas -Aquinas,  126 

Thome,  L.  W.,  387;  390 

Thompson,  S.  P.,  271,  334,  473 

Thomson,  J.,  463;  472,  484,  486 

Thomson,  J.  J.,  461,  474 

Thomson,  W.    See  Kelvin,  Lord 

Three  bodies,  problem  of,  240,  243,  252,  452- 
455;  Reduced  problem  of,  452 

Thybaut,  A.  L.,  325 

Thymaridas,  59,  in 

Tichy,  A.,  481 

Tides,  240,  264,  393,  336,  378,  449 

Timaeus  of  Locri,  25 

Time  a  fourth  dimension,  306,  480 

Timerding,  H.  E.,  480 

Tisserand,  F.  F.,  3S8,  455 

Todhunter,  I.,  370;  170,  171,  223,  230,  243- 
245,  331,  371,  437,  449,  465 

Tohoku  Mathematical  Journal,  82 

Ponelli,  A.,  423 

Tonstall,  C.,  146 

Top,  theory  of,  458 

Torricelli,  E.,  162,  163;  146,  156,  190 

Torsor,  335 

Tortolini,  B.,  346 

Tractrix,  1S2,  328 

Trajectories,  orthogonal,  217,  222,  322 

Transcendental  numbers,  2,  143,  362,  440, 
446,  447 

Transfinite  numbers,  426 

Transformation,  birational,  295,  314,  316, 
317,  319;  Linear,  295;  297 

Transon,  A.,  321 

Treviso  arithmetic,  128 

Triangle,  16,  18,  19,  71,  116,  297-300; 
Arithmetical,  76,  183,  187;  Right,  10, 
49,  56,  66,  71,  86,  104,  160.  165,  166; 
Similar,  16,  73;  Spherical,  46,  48,  50; 
Isosceles,  86,  104;  Heron's  formula  for 
area,  43,  66,  86,  123 

Triangular  numbers,  56,  168,  173 

Triangulum  characteristicum,  189,  207 

Trigonometry,  108,  109,  127,  131,  138,  149- 
157,  169,  222,  226,  229,  234-236,  265, 
483,  484;  Arabic,  104-106;  First  use  of 
word,  132;  Greek,  43,  47;  Hindu,  83,  96, 
97;  Notation  for  trig,  functions,  158; 
Notation  for  inverse  functions,  223; 
Spherical,  47,  76,  97,  105,  109,  no,  132, 
144,  267,  437,  481;  Trigonometric  func- 
tions,  cosecant,    106;   Cosirie,    no,    151; 


INDEX 


5x3 


Cota,.gent,  105,  xSt:  Tangent    104.  106. 
i32;Secant.  106,  132;  Sine.  u6.9Q,'o,. 

lo-;   no   151,  132;  Versed  sine,  q6 
rrisection  of '  an  angle.  2,  ^o.  36.  42,  104. 
100,  138.  142.  177.  202.  246;  Proved  im 
possible,  350 
Trisectrix,  229 
Tropfke.  J.,  211 
Trouton,  F.  T.,  47' 
Tnidi,  N.,  341 
Trucl,  H.  D.,  26s 
Tschebytschew.    .9<-c  Chebichev 
Tschimhausen,  E.  W     225:  209.  2x0    226 
253,  349;   H's  transformation,   225,  349 
Tsu  c'h'  ung'Chih,  73  . 

Tucker,  R.,  299;  414;  His  circle,  300 
Two  mean  proportionals,  19 
Tycho  Brahe,  105,  iS9 
Tzitzeica,  G.,  31S 
Ubaldo,  G.,  172 

Ste'raUos.    See  Prime  and  ultimate  r. 
SSi^.^:;Sients.     .eelndetermi- 

nate  c. 
Unger,  F.,  128 
Unifomiization,  433 
Universities,  math's  in,  129 
Unknown  quantity,  symbol  for,  75 
Vacca,  G.,  142 
Vahlen,  T.,  327,  446 
Vailati,  G.,  328 
Valentin,  G.,  42S 
Valentiner,  H.,  359,  360 
VaUee  Poussin,  C.  J.  de  la,  376,  439 
Valson,  C.  A.,  368 
Van  Cculen,  L.,  i43 
Vandermonde,  C.  A.,  266;  253,  264 
Vandiver,  H.  S.,  443 
Van  Orstrand,  C.  E.,  484 
Van  Schooten.    See  Schooten 
Van  Velzer,  C.  A.,  341  ^         .        a 

Van  Vleck,  E.  B.,  40S,  406;  Quoted,  396, 

405,  422 
Varaha.  Mihira,  84 
Variable  parameters,  211 
Variables,  complex,  420 
Variarions.   calculus  of.     See   Calculus  of 

variations 
Variation  of  arbitrary  constants,  240 
Varignon,  P.,  224;  iS6,  185,  220,  222 
Varicdk,  V.,  481 
Vasiliev,  A.,  425 
Vavasseur,  R.  P-  Le.  357,  360 
Veblen,  ()..  30g.  324.  326.  328,  409 
Vector  analysis,  308,  322.  334,  335.  452,  470 
Vega,  G.  V...  ist>.  482 


Velaria,  221 

Vene,  A.  A..  270 

Venn,  J.,  379;  407,  4o8 

Vcnturi,  44 

Venturi,  A.,  452 

Verbiest,  F.,  77 

Vernier,  P.,  142 

Veronese,  G.,  291,  307,  309,  327 

Vibrating  strings,  226,  242,  251,  252.  258, 

464 
Vicat.L.  J.,  467:468 
Victorius,  65 

Vieta,  F.,  137-139;  41,  I".  133,  141,  142, 
143, 144, 146. 156, 174.  177.  178.  187.  192, 
203,  253 
Vigesimal  system,  69,  70 
Vija-ganita,  85 
Villarceau,  A.  Y.,  452;  482 
Vincent,  A.  J.  H.,  33° 
Vincent,  Gregory  St.,  181 
Vinci,  Leonardo  da,  273 
Virtual  velocities,  2q,  172,  255 
Vitali,  G.,  405 
Vivanti,  G.,  218,  427 
Viviani,  G.,  409 
Viviani,  V.,  162 

Vlack,  A  ,  151;  77,  152,  i54,  4»2 
Vogt,  H.,  29 
Vogt,  W.,  308 
Voigt,  W.,  471,474 
Volpi,  R.,  40s 
Volterra,  V.,  346,  372,  387,  393.  394,  39.S, 

40s,  406 
Von  Staudt,  280,  395.  409,  436 
Voss,  A.,  297,  308,  325,  374,  394;  Quoted, 

410 
Xenocrates,  26 
Xylander,  141 
Wachter,  F.  L.,  30S 
Wada,  Nei,  81 
Wagner,  U.,  128 
Waldo,  F.,  463 
Wallenberg,  G.  F.,  387 

S"f':^-^S8-88,xo8.x37,X46.u8. 

''     t'/5,x58,x65,x68,x69.x78.x79.i8x 

,go,  191.  192,  X96,  202,  213,  2X5,  235,  265. 

302,  iob,  331,  343,  373 
Wallner,  C.  R.,  126 
Waltershausen,  W.  S.,  232 
Walton.  J.,  219 
Walton,  W.,  324 
Wand,  Th.,  476 
Wang  Hs'  lao-t'ung,  74 
Wantzel,  P.  L.,  35°;  4^6 
Ward,  Seth,  iS7  ,.,    ,^8. 

V^aring,  E.,   248-249;   143,   202,   241.  24». 


514 


INDEX 


2S4<  320,  361,  364;  Miscellanea  analytica, 
241;  Waring's  theorem,  248,  442,  443 
Warren,  J.,  330 
Watson,  J.  C,  455 
Watson,  S.,  379 

Watt,  J.,  300;  Watt's  curve,  300 
Wave  theory  of  light,  183 
Weaver,  J.  H.,  50 

Weber,  H.,  361;  318,  353,  357,  sqq,  400,  418 
Weber,  W.,  6,  421,  434,  467,  474 
Weddle,  Th.,  365;   155,  319;  Surface,  310 
Weierstrass,  K.,  423-426;  32,  258,  270,  285; 
326,  345,  346,  347,  362,  368,  370,  371,  372, 
376,  385,  388,  397,  398,  399,  400,  415, 
417,  418,  422,  428,  429,  430,  431,  432, 
446,  453,  456;  Weierstrass'  Construction, 
372 
Weigel,  E.,  205 
Weiler,  A.,  384 
Weingarten,  J.,  314;  325 
Weir,  P.,  482 
Weissenborn,  H.,  115 
Weldon,  W.  F.  R.,  381 
Wendt,  E.,  357 
Werner,  J.,  141 
Werner,  H.,  347 
Wertheim,  G.,  60 
Wertheim,  W.,  468 
Wessel  C,  265;  420 
Westergaard,  H.,  380,  381 
Wetli,  486 

Weyl,  H.,  391,  465,469 
Wheatstone,  C,  465 
Whewell,  W.,  324;  37,  160,  240 
Whipple,  J.  W.,  48s 
Whist,  383 
Whiston,  W.,  201 

White,  H.  S.,  278,  300;  Quoted  3,  450,  295 
Whitehead,  A.  N.,  407, 409;  Quoted,  294, 328 
Whitley,;.,  298 
\\-hitney,  W.  D.,  85 
Whittaker,  E.  T.,  386 
Widmann,  J.,  139;  125 
VVieferich,  A.,  443 

Wieleitner,  H.,  127,  174,  182,  235,  250 
Wiener,  A.,  366 

Wiener,  C,  297;  274,  276,  317,  326 
Wiener,  H.,  289,  329 
Wiener,  N  ,  409 
Wilczynski,  E.  J.,  322 
Williams,  K.  P.,  392 
Williams,  T.,  265 

Wilson,  E.  B.,  335,  401,  481;  Quoted,  327 
Wilson,  J.,  248;  254;  Wilson's  theorem,  248, 

254 
Winckler,  A.,  469 
Wing,  \' ,  157 


Wingate.  E.,  481 
Winlock,  J..  383 
Winter,  M.,  410 
Witt,  F.  de.  See  De  Witt 
Wicting,  A.,  318 
Wittstein,  A.,  291 

Wittstein,  T.,  381 

Woepcke,  68,  100 

Wolf,  C,  158,  175,  226 

Wolf,  R.,  259,  379 

WoIfBng,  E.,  324 

Wolfram,  247 

Wolfshekl,  F.  P.,  443 

Wolstenholme,  J.,  379 

Woodhouse,  R.,  272;  219,  370 

Woodward,  R.  S.,  459 

Woolhouse,  W.  S.  B.,  365,  379 

Wren,  C,  166;  179,  181,  188,  199,  275 

Wright,  E.,  153,  155,  180 

Wright,  J.  E.,  356 

Wright,  T.,  451 

Wronski,  H.,  340;   258;  Wronskians,  34c 

Yan  Hui,  75 

Yendan  method,  80 

Yenri,  80,  81 

Yoshida  Shichibei  Koyti,  78;  79 

Young,  A.,  348 

Young,  G.  C,  3?5,  326 

Young,  J.  R.,  271 

Young,  J.  W.,  328 

Young,  Th.,  470;  n,  183.  464,  465 

Young,  W.  H.,  325,  326,  406 

Yii,  emperor,  76 

Yule,  G.  v.,  381 

Zach,  484 

Zehfuss,  G.,  341 

Zeller,  C.,  436 

Zeno  of  Elea,  23;  24,  29,  51;  On  motion, 

48,  67,  126,  182,  219,  400 
Zenodarus,  42;37o 
Zermelo,  E.,  372,  401, 402,  403;  Principle  of, 

401 
Zero,  invention  and  use  of,  119,  121,  147; 
2,  5,  116;  by  Maya,  69;  Symbols  for,  5, 
S3.  69,  75,  78,  88,  89,  94,  100;  division  by, 
94,  284 
Zero-denominator,  185 
Zero,  first  use  of  term,  128 
Zerr,  G.  B.  M.,  379 
Zeuner,  G.,  381 

Zeuthen,  H.  G.,  32,  190,  293,  314,  316^  320 
Zeuxippus,  34 
Zizek,  F.,  380 
ZoUner,  F.,  309 
Zolotarev,  G.,  442,  444 
Zorawski,  K.,  356 
Zyklugnphie,  297 


ADDITIONS  AND  CORRECTIONS 

tP.  =  page,  1.  =  line,  F:  =  for,  R:  =  read.) 

P.  4  1.  13,  F:  employed  R:  employed  in  most  instances; 
P.  4  1.  35,  F  :  posses  R:  cite;  P.  10  1.  19,  F:  sides  R:  legs; 
P.  II  1.  I,  F  :  of  3000  R  :  of  perhaps  3000 ;  P.  19  1.  3,  F  :  solids 
R :  regular  solids  ;  P.  23  11.  28,  36,  F :  paradoxies  R  :  paradoxes  ; 

P.  32  1.  13,  F  :  ^  R :    =  ;  P.  36  1.  37,  F :  Appolonius  R  :  Apol- 

lonius ;  P.  39  1.  21,  F :  sides  R :  elements  ;  P.  49  1.  30,  F :  of  a 
R :  of  the  area  bounded  by  a ;  P.  52  1.  39,  F  :  zeta  R :  vau ; 
P.  53  1.  4,  F  :  final  sigma  R  :  vau  ;  P.  56  1.  14,  F  :  called  R  :  called 
by  the  later  Greeks  and  Euclid ;  P.  57,  1.  42,  F :  Mittelalter 
und  Alterthum  R :  Alterthum  und  Mittelalter ;  P.  58,  1.  6,  F : 
are  R:  and  for  perfect  numbers  are;  P.  60  1.  21,  F:  age  was 
R :  age  at  the  time  of  his  death  was ;  P.  80,  1.  34,  F :  in  R : 
known  in  ;  P.  80  1.  38,  F :  n'  R :  nl;  P.  93  1.  29,  F :  has  R : 
(having  real  roots)  has  ;  P.  99  1.  20,  F  :  Euphrates  R  :  Tigris  ; 

P.  105  1.  22,  F :  V  1  +  ^'  R  :  V  (1  +  -0') ;  P-  105  1-  27,  F  :  or 
R  :  of  ;  P.  106  1.  25,  F:  discovered  R:  observed:  P.  114  1.  13 
F :  Ireland  R  :  York  ;  P.  1 16  1.  14,  F :  Fine  R :  Fine  ;  P.  1 16 1.  37' 
dele  :  a  ;  P.  122  1.  31,  F  :  nor  R  :  not ;  P.  123  1.  36,  F :  Tivolis' 
R  :  Tivoli's  ;  P.  127  1.  2,  F  :  profundis  R  :  profundus  ;  P.  133 1.  39, 

F :  jc'  =  mx  —  ?i  R :  x'  +  mx  =  n  ;  P.  133  1.  41, 


.:(«)■  K:(|;F 


R  :  (  —  ]  ;  P.  136  1.  13,  F  :  equations  R  :  general  equations  ; 

P.  137  1.  46,  F:  Trigonometry  R:  Trigonometric  ;  P.  139  11.  13, 
14,  in  the  quotation,  F  :  a,  6  R  :  A,  B,  respectively  ;  P.  142  1.  46, 
F:  Coimpre  R:  Coimbre ;  P.  156  1.  7,  F:  Wallis  in  1695  ob- 
tained ^  log(l  +  2)/log(l  —  z)  R :  James  Gregory  in  1668  and 
Edmund  Halley  in  1695  obtained  substantially  the  series 
I  log(l  +z)—  ^  log(l  —  2;)  ;  P.  156  1.  20,  F :  Pierre  Varignon  •  • 
1722  R:  Jakob  Bernoulli  in  a  paper  published  in  the  Acta  eru- 
ditorum  in  1691 ;  P.  157 1.  2,  F :  equations  R :  equations  (having 
515 


516  A    HISTORY    OF    MATHEMATICS. 

real  roots)  ;  P.  157  1.  5,  F:  in  an  R:  in  such  an;  P.  167  1.  38, 
F :  23550336  R :  33550336 ;  P.  167 1.  43,  F :  by  R  :  by  J.  Pervusin 
and;    P.    181,    1.    8,    F:   hyperbola   back   R:   parabola   back; 
P.  181  1.  8,  F :  The  curve  E :  The  earliest  absolute  rectification 
was  that  of  the  logarithmic  spiral,  by  E.  Torricelli  in  1640. 
The   curve;  P.  181  1.  10,  F:  first  E:  next;  P.  182  1.  3,  dele: 
possibly;  P.  184  1.  24,  F :  C.  S.  Clavio  E:  C.  Clavius ;  P.  184 
1.  265  dele  :  The  existence  ■  •  Simplicius  ;  P.  185  1.  29,  F :  greater 
than  unity  and  negative  E  :  less  than  minus  one  ;  P.-  209  1.  34, 
F :  Berlin  E  :  Leipzig  ;  P.  213  1.  30,  F  :  G.  G.  E  :  G.  W. ;  P.  239 
1.  23,  F  :  3  E :  2  pairs  were  false,  3  ;  P.  239  1.  26,  F  :  2  n  E  :  2"  ; 
P.  240  1,  25,  also  P.  244  1.  26,  P.  477  1.  24,  P.  503  1.  13.  F :  Mau- 
pertius  E:  Maupertuis  ;  P.  250  1.  22,  F  :  ,  but  E  :  ;  a  related 
curve  was;  P.  271   1.  35,  F:  3765  E:  1865;   P.  282  1.  8,  F: 
them  E:  it;  P.  287  1.  3,  F  :  Synthetic  E:  Modern  synthetic; 
P.  296  1.  6,  F  :  at  E :  out  at ;  P.  299,  in  the  figure,  Z  ABO  =  o, 
Z  ACD  =  &  ;  P.  310  1.  2,  F :  abbreviated  E :  abridged  ;  P.  310 1.  4, 
F :  Marne]  E :  ^Vlarne,  by  G.  Lame  and  by  J.  D.  Gergonne]  ; 
P.  312  1.  28,  F:  Flachen.  E  :  Flachen ;  P.  316  1.  46,  F:  H.  B. 
R:  H.  F. ;  P.  320  1.  28,  add:  It  was  enlarged  and  translated 
into   French   in   1908   and   1909;  P.  320  1.  34,  F:E.P.  C.  E 
F.P.C. ;  P.  323  1.38,  F:  Peterson  E:  Petersen;  P.  327  1.  38. 
F:    sum    E:   angle    sum;  P.  330  1.  10,  F :  e^'*'^*"  =   E:e2""r'  = 
P.  338  1.  18,  dele  :  first ;  P.  348  11.  34,  35,  F  :  F.  W.  P.    E  :  T. 
P.  353  1.  40,  F  :  1861  E  :  1860  ;  P.  358  1.  46,  F  :  only  one  E :  no 
P.  360  1.  33,  F  :  degree  E  :  order  ;  P.  361  1.  26,  F  :  was  E  :  were 
P.  372  1.  39,  F  :  1862  E  :  1862-1867  ;  P.  373  1.  3,  F :  this  E 
the  preceding;  P.  373  1.  17,  F:  thus  named  E:  bearing  a  name 
used;  P.  401  1.  32,  F:  V.  A.  E :  H. ;  P.  429  1.  17,  F :  Dirichlet's 
R :   Thomson-Dirichlet's ;  P.  444  1,  39,  F :  first  step  R :  step 
P.  489  1.  37,  F  :  1758  E  :  1759  ;  P.  494  1.  42,  F  :  Emsh  E  :  Emch 
P.  499  1.  44,  F :  169,  331  E :  142,  169,  331 ;  P.  502  11.  3,  4,  dele 
401,  402,  405,  406,  421 ;  P.  502  1.  4,  insert :  Lebesgue,  H.,  406 
401,  402, 406, 421 ;  P.  503  1.  25,  dele :  360 ;  P.  503  1.  47,  F  :  142 
R:  142,  169,  331 ;  P.  507  1.  11,  F:  Peterson  R:  Petersen. 


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